The Limit Points of the Commuting Probability Function on Finite Groups

The Limit Points of The Commuting Probability Function on Finite Groups

Madeleine Whybrow, CID 00640867

Supervisor: Dr John Britnell

June 2014

This is my own unaided work unless stated otherwise. Abstract

If G is a ﬁnite group, then Pr(G) denotes the probability that two elements of G, chosen at random, commute. Joseph [15] has conjectured that if l is a limit point of the function Pr on ﬁnite groups then l is rational and is not approached from below. Hegarty [12] showed that this is true if l ∈ (2/9, 1]. We discuss the various steps in Hegarty’s proof, as well as suggesting ways to extend his work and prove Joseph’s conjectures over a larger range. Acknowledgements

I would like to thank my supervisor Dr John Britnell for his support and en- couragement. His dedication and expertise has made the writing of this project an enjoyable and educational experience. Contents

1 Introduction 2

2 Preliminaries 5 2.1 Commutators and the Commutator Subgroup ...... 5 2.2 Restrictions on the Commutator Subgroup ...... 6 2.2.1 Complete Groups ...... 6 2.2.2 Dihedral Groups ...... 8 2.2.3 Restrictions on Z(G)...... 10 2.3 Nilpotent Groups ...... 10 2.4 Isoclinic Groups ...... 11

3 Key Results on the Commuting Probability 15 3.1 Basic Results ...... 15 3.2 The Commuting Probability and Character Degrees ...... 18 3.3 Commuting Probability of Semigroups ...... 20

4 Background 22 4.1 Bounds on The Commuting Probability ...... 22 4.2 Classification Results ...... 24 4.2.1 Classification of Groups with Pr(G) > 11/32 ...... 24 4.2.2 Classification of Groups of Odd Order with Pr(G) > 11/75 29

5 Joseph’s Conjectures 33 5.1 Hegarty’s Work ...... 33 5.1.1 Groups with |G0| < 8...... 33 5.1.2 The Main Proof ...... 35 5.2 Further Work ...... 37 5.2.1 Groups with |G0| = 8...... 37 5.2.2 A General Formula for the Case |G0| ≤ n...... 38

6 Concluding Remarks 41

1 Chapter 1

Introduction

Let G be the set of all ﬁnite groups then, for G ∈ G, we deﬁne Pr(G) to be the probability that two elements selected at random (with replacement) from G commute with each other. That is to say

|{(x, y) ∈ G × G : xy = yx}| Pr(G) = . |G|2 The commuting probability of a group can be thought of as a measure of how commutative a group is. In fact, some sources call Pr(G) the commutativity degree of G to draw attention to this. Many of the results on the probabilty take the approach of linking this numerical measure of commutativity with some structural indicator such as the size of the commutator subgroup or the nilpotency class of the group. During the second half of the twentieth century, there was a spate of work studying the commuting probability. The majority concentrated either on bounding Pr for certain groups or on classifying groups with certain values of Pr. We give an overview of the most important results in Chapter 4. In this paper, we will take a slightly diﬀerent approach, treating Pr as a function from G to (0, 1]. In particular, we will study the limit points1 of the image Pr(G). To our knowledge, this perspective had not been seriously considered until Keith Joseph [15] published a note in the American Mathematical Monthly proposing the following three conjectures:

1. Every limit point of Pr(G) is rational.

2. If l is a limit point of Pr(G), then there exists = l > 0 such that Pr(G) ∩ (l − l, l) = φ.

3. Pr(G) ∪ {0} is a closed subset of R.

1 For our purpose, we say that l ∈ R is a limit point of a set S ⊆ R if there exists a sequence (sn) of elements of S such that sn → l.

2 For brevity, we say that a subset S ⊆ R is a good set if, for every limit point l of S, the following hold:

1. l ∈ Q

2. there exists = l such that S ∩ (l − l, l) = φ.

Note also that any subset of a ﬁnite union of good sets is also good. In his orginal article, Joseph gives little mention of the intuition behind these conjectures. However, the following result gives some indication to the motivation behind the ﬁrst two conjectures.

Lemma 1.1 ([12]). Let n ∈ N be ﬁxed then

( n ) X 1 S := : x ∈ n x i N i=1 i is a good set. So in order to prove Joseph’s conjectues, it would be sufficient to show that G consists of a finite number of families of groups whose commuting probabilities are contained in Sn for some n fixed. There are a number of results (see for example Section 4.2) which show that certain families of groups do indeed have commuting probabilities of this form. Some would have been available at the time when Joseph published his conjectures and we would guess that it is these which motivated his conjectures. Joseph’s conjectures were left untreated, as far as we are aware, until recently Hegarty [12] proved that the first two conjectures hold for limit points in the 2 range ( 9 , 1]. He gives no attention to the third conjecture, save for commenting that it appears much more mysterious than the first two. His method relied on the following result, proved by Guralnick and Robinson [9] which links the commuting probability with the size of the commutator subgroup, G0 (see Section 2.1), and the smallest degree of an irreducible character of G. The proof can be found in Section 3.2. Theorem 1.2. If G is non-abelian, then

1 1 1 Pr(G) ≤ + 1 − d2 d2 |G0| where d is the smallest degree of a non-linear complex irreducible character of G. In fact, when d = 3 and |G0| = 8,

1 1 1 2 + 1 − = . d2 d2 |G0| 9 2 0 So if Pr(G) > 9 then either |G | < 8 or d = 2 (as the character is non-linear). The rest of the proof is then split between these cases.

3 We aim to improve his method, working towards proving Joseph’s ﬁrst two conjectures over a larger range. In particular, ( 1 1 1 17 ≈ 0.2099 if d = 3, |G0| = 9 + 1 − = 81 d2 d2 |G0| 23 0 128 ≈ 0.1797 if d = 4, |G | = 8. So, in order to prove the conjectures over a wider range, we must ﬁrst prove that the conjectures are true when |G0| < 9. The case |G0| < 8 is treated by Hegarty but only in brief detail. Thus, a large part of our work involves describing explicitly his method. We then discuss possible methods to extend his work.

4 Chapter 2

Preliminaries

2.1 Commutators and the Commutator Subgroup

Deﬁnition 2.1. For a group G, the commutator of two elements x and y in G is [x, y] = x−1y−1xy. Equally, if X and Y are two subsets of G then

[X,Y ] = h[x, y]: x ∈ X, y ∈ Y i. In particular, the group [G, G] is denoted G0 and is called the commutator or derived subgroup.

It is important to note that the product of two commutators is not necessarily a commutator, that is to say that the set of commutators may not be a group. Isaacs [14] constructs a source of examples of groups where this is the case. Coversely, the conjugate of a commutator by any element of G lies in G0 as

[x, y]g = [xg, yg].

0 This clearly shows that G E G. However, a stronger result is that, for φ a group endomorphism of G, we have

φ([x, y]) = [φ(x), φ(y)]. That is to say, G0 is invariant under every endomorphism of G, or that it is a fully characteristic subgroup of G. This property is considerably stronger than normality. Another useful property of the commutator subgroup is its relationship to abelian quotient groups:

Proposition 2.2. If G is a group and N E G then G/N is abelian if and only if G0 ⊆ N.

5 Proof. Suppose for contradiction that there exists N E G such that G/N is abelian and that there exists x, y ∈ G such that [x, y] ∈/ N. Then, working in G/N we have

[xN, yN] = xNyNx−1Ny−1N = [x, y]N 6= N.

So [xN, yN] 6= eG/N and xN and yN do not commute. Thus, if G/N is abelian, all commutators, and by extension the commutator subgroup, lie in N. Conversely, if G0 ⊆ N then ∀x, y ∈ G,[x, y] ∈ N. In particular, [xN, yN] = eG/N and so G/N is abelian.

An immediate consequence of this result is that G0 is the unique minimum subgroup of G such that its quotient is abelian. As such, G/G0 is called the abelianisation of G. We also note the following result on the centraliser of the commutator subgroup: Lemma 2.3. For any group G

0 0 0 (CG(G )) ⊆ Z(G) ⊆ Z(CG(G )). Proof. This follows from the Hall-Witt identity:

[x, [y−1, z]]y[y, [z−1, x]]z[z, [x−1, y]]x = e.

2.2 Restrictions on the Commutator Subgroup

For a given group G, it is not always true that there exists a group H such that H0 =∼ G. In certain cases, the commutator subgroup will also limit the groups which Z(G) can be isomorphic to, as shown in Section 2.2.3. The results in this section will be used to rule out certain possibilities in later proofs. For completeness, we have chosen to prove results which are more general than those we require.

2.2.1 Complete Groups Here we identify a class of groups which may not be commutator subgroups. The most notable groups of this form are the symmetric groups Sn for n 6= 6, n ≥ 3. Deﬁnition 2.4. A group G is complete if Z(G) = {1} and Aut(G) = Inn(G). Note that the map φ : G → Aut(G) sending the element g to conjugation by g is injective if and only if Z(G) = {1} and is surjective if and only if Aut(G) = Inn(G). Thus, a complete group is one where the map φ is an isomorphism.

6 Proposition 2.5. If G is complete and G0 6= G then G is not isomorphic to the commutator subgroup of any other group. Proof. Suppose there exists H such that H0 = G. As G is normal in H, h ∈ H induces an automorphism of G by conjugation. However, Inn(G) = Aut(G) so this conjugation must be equal to conjugation by some element, say g, in G. This means that conjugation by g−1h must be the −1 identity on G and so is contained in CH (G). Thus, we may write h = g(g h) ∈ GCH (G). Moreover, as Z(G) = {e}, CH (G) ∩ G = {e} so H = G × CH (G). 0 From Lemma 2.3, (CH (G)) ⊆ Z(H) ⊆ Z(G) = {e} and CH (G) is abelian. 0 0 0 0 0 Thus H = G × CH (G) = G , contradicting the assumption that G 6= G .

We will now show that the symmetric group Sn is complete for n 6= 6, n ≥ 3. To begin, we note that the automorphism group of a group G acts on the −1 conjugacy classes of G. That is, if α ∈ Aut(G) and if ClG(x) = {gxg : g ∈ G} −1 then α(ClG(x)) = {α(g)α(x)α(g ): g ∈ G} is also a conjugacy class.

Lemma 2.6. If α ∈ Aut(Sn) stabilises the conjugacy class of transpositions then α ∈ Inn(Sn).

Proof. We want to show that there exists an element σ ∈ Sn such that for all g,

α(g) = σgσ−1. (2.1)

In fact, since every non-identity element of Sn is the product of transpositions of the form (1 i) for 1 < i ≤ n, we need only show (2.1) is true for g of this form. Let 1 < l1 < l2 ≤ n then, as (1 l2 l1) = (1 l1)(1 l2) has order 3, α((1 l1))α((1 l2)) does too. Thus α((1 l1)) and α((1 l2)) are of the form (a b) and (a c) for some a, b, c. As we chose l1 and l2 distinct, this a is independent of the transpositions used. We may deﬁne a function f mapping 1 to a and any 1 < k ≤ n to the image of 1 under the permutation α((1 k)). If we let σ be the permutation sending 1 to a and k to f(k) then (2.1) is satisﬁed. Lemma 2.7. For n 6= 6, the class of transpositions is the unique class of n involutions with 2 elements. Proof. Suppose we have a class of involutions which are the product of k trans- n positions and that this class has size 2 . Then 1 nn − 2 n − 2(k + 1) n ... = . k! 2 2 2 2 Rearranging and setting m = n − 2 and k = r + 1 gives

m m − 2(r − 1) ... = (r + 1)!. 2 2 However, the left hand side can be rewritten as

7 m(m − 1) ... (m + 2 − 2r)(m + 1 − 2r) m! = . 2k (m − 2r)2r So 2.2.1 becomes

m (r + 1)!2r = 2r (2r)! For r = 1 this is m(m − 1) = 4 which has no integral solutions. For r = 2 this is m(m − 1)(m − 2)(m − 3)(m − 4) = 24 which as one integral solution; m = 4 (or equivalently n = 6). For r > 2 the right hand side is less than 1 so is not a integer.

Proposition 2.8. For n 6= 6, Aut(Sn) = Inn(Sn). Proof. The automorphisms permute conjugacy classes and when n 6= 6 must, in particular, stabilise the class of transpositions as it is the only class of its size. Thus, from Lemma 2.6, all automorphisms are inner.

Proposition 2.9. For n ≥ 3, Z(Sn) = {e}.

Proof. Choose π ∈ Sn such that π 6= e, then there exists i such that π(i) = j 6= i. As n ≥ 3, there exists ρ = (j k) ∈ Sn such that k 6= i, j. In particular, ρ ﬁxes i, as does ρ−1 so

ρπρ−1(i) = ρπ(i) = ρ(j) = k 6= j.

−1 Thus ρπρ 6= π and Z(Sn) = {e}.

These two propositions show that if n ≥ 3 and n 6= 6, then Sn is complete.

2.2.2 Dihedral Groups Here we identify another class of groups which may not be commutator subgroups. The most notable elements of this class are the dihedral groups. We first recall the following definition: Definition 2.10. A subgroup H of G is characteristic if φ(H) = H for every automorphism φ of G. We write “H is a characteristic subgroup of G” as H char G. The proof of the following result can be found on p. 104 of [24]: Proposition 2.11. Let G be a group.

1. If N E G and X char N then X E G. 2. If N char G and X char N then X char G.

8 Proposition 2.12. If G is a group with a characteristic subgroup, K, such that Aut(K) is abelian and K 6⊆ Z(G), then G is not isomorphic to the commutator subgroup of any group. Proof. Suppose there exists H such that H0 = G. Since K is characteristic in G and G is normal in H, K is normal in H. Thus there is a homomorphism, φ, from H to Aut(K) mapping g ∈ H to conjugation by g. However, Aut(K) is abelian so G/Ker(φ) = Im(φ) is too and G = H0 ⊂ Ker(φ). This is equivalent to saying that K is contained in the centre of G, giving a contradiction with the initial assumptions.

We now prove that the dihedral group D2n for n ≥ 3 obeys the conditions in Proposition 2.12. Recall that the dihedral group is deﬁned as the group with presentation

ha, b | a2 = e, b2n = e, ba = ab−1i. and that for all r ∈ N

bra = ab−r. Lemma 2.13.  D if n = 1, 2,  2n Z(D2n) = {e} if n > 2 is odd , {e, bn/2} if n > 2 is even .

Proof. If n = 1 or 2 then D2n is abelian so Z(D2n) = D2n. Since a and b generate D2n, x ∈ Z(D2n) if and only if it commutes with a and b. If we write x = aibj then x commutes with b if and only if aibj+1 = baibj, that is, aib = bai. If n ≥ 3, the only value of i for which this is true is i = 0 giving x = bj. Thus x commutes with a if and only if bja = abj, that is, if and only if ab−j = abj. So we have b2j = e. Since o(b) = n, this give n|2j. However, as 0 ≤ j ≤ n − 1, either j = 0 or 2j = n. If j = 0 then x = e. If 2j = n, n is even and x = bn/2.

Lemma 2.14. The subgroup K = hbi ≤ D2n is a characteristic subgroup such that Aut(K) is abelian. Proof. Let x = abi where i ≥ 0. Then x2 = (abi)(b−ia) = e so is of order 2. Thus K is the only subgroup of order 2n and so is characteristic by uniqueness. Furthermore, as K is cyclic, if φ, ψ ∈ Aut(K) then for all x ∈ K, φ : x 7→ xs and ψ : x 7→ xt where hcf(s, n) = hcf(t, n) = 1. Then ∀x ∈ K

φ ◦ ψ(x) = (xt)s = xts = xst = (xs)t = ψ ◦ φ(x). Hence Aut(K) is abelian, as required.

9 2.2.3 Restrictions on Z(G) When considering the commuting probability of a group, it is often useful to study the intersection of its commutator subgroup and its centre. The following Lemma can be inferred directly from the definition of the centre. Lemma 2.15. If there exists x ∈ G0 such that x is fixed by all inner auto- mophisms of G, then x ∈ Z(G). 0 ∼ 0 In particular, if G = C2n for n ≥ 1 then G contains a unique element of order 2. As it is unique and G0 is normal in G, it must be fixed by all inner automorphisms of G, so is central in G. Thus Z(G) must be of even order.

2.3 Nilpotent Groups

Nilpotent groups feature heavily in our method. In this section, we will recall some of the key deﬁnitions and results on the topic. See, for example, Rotman (1995, pp. 112-7) for a complete introduction to nilpotent groups and for the omitted proofs in this section. Deﬁnition 2.16. A central series of G is a sequence of subgroups

{e} = A0 /A1 /.../An = G

such that [G, Ai+1] ≤ Ai for 0 ≤ i ≤ n. Not every group has a central series, those which do are called nilpotent. The minimal length of a central series of a nilpotent group is called its nilpotency class. Deﬁnition 2.17. The series of subgroups

G = G1 D G2 D ... D Gn+1 D ...

such that Gi+1 = [Gi,G] for i ≥ 1 is called the lower central series of G. We sometimes write Gi(G) if we wish to draw attention to the group in question. Deﬁnition 2.18. The sequence of subgroups

{e} = Z0 E Z1 E ... E Zn E ...

such that Zi+1 = {x ∈ G : ∀y ∈ G, [x, y] ∈ Zi} is called the upper central series of G. We sometimes write Zi(G) if we wish to draw attention to the group in question. Every group possesses a lower and an upper central series but they are ﬁnite if and only if the group is nilpotent. Of particular importance is the relationship between nilpotent groups and p-groups. Every p-group is nilpotent and as a partial converse: Theorem 2.19. If G is nilpotent then G is equal to the direct product of its Sylow p-subgroups.

10 Proposition 2.20. If G and H are groups such that θ : G → H is a surjective homomorphism then θ(Gn) = Hn for n ≥ 1. In particular if we take H = G/Z(G) and θ to be the quotient map then we have the following result:

Corollary 2.21. For n ≥ 2

G G Z(G) =∼ n . Z(G) n Z(G) 2.4 Isoclinic Groups

The concept of isoclinic groups was first introduced by Hall [11]. In his 1940 paper, he studies the classification of finite p-groups. Instead of using the equivalence classes of isomorphic groups for his classification, he uses the equivalence relation of isoclinism.

Deﬁnition 2.22. Two groups G1 and G2 are isoclinic if there exist isomorphisms

G1 G2 0 0 φ : → , ψ : G1 → G1 Z(G1) Z(G2) such that the following diagram commutes:

φ×φ G1 × G1 −−−−→ G2 × G2 Z(G1) Z(G1) Z(G2) Z(G2)   ρ1 ρ2 y y 0 ψ 0 G1 −−−−→ G2

0 where ρi : Gi/Z(Gi)×Gi/Z(Gi) → Gi is deﬁned as ρi :(xZ(Gi), yZ(Gi)) 7→ [x, y] for i = 1, 2. It is easy to show that this forms an equivalence relation on the set of all ﬁnite groups. It is weaker than isomorphism but preserves some important group properties. In particular, Lescot [19] showed that if two groups are isoclinic then they have the same commuting probability (proved below). He used this fact to classify groups such that Pr(G) ≥ 1/2. Proposition 2.23. If G and H are isoclinic then Pr(G) = Pr(H).

Proof. Suppose that G1 and G2 are isoclinic with isomorphisms φ and ψ as in the deﬁnition above. Then

11 2 G1 1 2 Pr(G1) = 2 |G1| Pr(G1) Z(G1) |Z(G1)| 1 = 2 |{(x, y) ∈ G1 × G1 : xy = yx}| |Z(G1)| 1 = 2 |{(x, y) ∈ G1 × G1 :[x, y] = 1}| |Z(G1)| 1 = 2 |{(x, y) ∈ G1 × G1 : ρ1(xZ(G1), yZ(G1)) = 1}| |Z(G1)| ( ) G 2 1 = (a, b) ∈ : ρ1(a, b) = 1 . Z(G1) As ψ is an isomorphism, this equals ( ) G 2 1 (a, b) ∈ : ψ(ρ1(a, b)) = 1 . Z(G1) By the commuting diagram, this equals ( ) G 2 1 (a, b) ∈ : ρ2(φ(a), φ(b)) = 1 . Z(G1) Finally, as φ is an isomorphism, this equals ( ) G 2 2 (c, d) ∈ : ρ2(c, d) = 1 . Z(G2) However, from above, ( ) G 2 G 2 2 2 Pr(G2) = (c, d) ∈ : ρ2(c, d) = 1 . Z(G2) Z(G2) Thus,

2 2 G1 G2 Pr(G1) = Pr(G2). Z(G1) Z(G2)

Which, as G1/Z(G1) and G2/Z(G2) are isomorphic and thus have the same size, implies Pr(G1) = Pr(G2).

The following is a classic result on isoclinic groups. It was ﬁrst stated by Hall [11] although he gives only an outline of the proof.

12 Proposition 2.24. If G is any finite group then there exists a group G1, iso- 0 clinic to G, such that Z(G1) ⊆ G1. A group such that Z(G) ⊆ G0 is known as a stem group. We outline a proof of this result found in Lescot [19]. We give the construction of the stem group G1 and the maps needed to show that it is isoclinic to G but omit the checks which show that G1 is indeed a stem group and that the maps are well-defined isomorphisms. Before we do so, we introduce some results about free groups: Definition 2.25. For S a given set, take S−1 to be the set of “inverses” of elements of S. That is to say, if u ∈ S then there exists an element u−1 of S−1. The free group FS is then the group comprising of all expressions (or words) built from elements of S ∪ S−1 omitting pairs of the form uu−1 for u ∈ S. The group operation in FS is concatenation of words and the identity is the empty word. An arbitrary group is free if it is isomorphic to FS for some subset S of G. If a group has a generating set S then there is a surjective homomorphism π : FS → G or, equivalently, G is isomorphic to a quotient group of the free group FS. Also, ker π is a set of relations in the presentation of G. A similar, but subtly different idea is that of free abelian groups.

Deﬁnition 2.26. A group is free abelian if it is isomorphic to the direct sum ∼ L of copies of Z, that is to say, G = s∈S Z for some set S. They are linked to free groups by the following result: Lemma 2.27. If F is a free group then F/F 0 is isomorphic to a free abelian group. In fact, free abelian groups may be equivalently deﬁned as the abelianisation of free groups. Intuitively, a free abelian group on a set S consists of words in the free group FS where two words are equal if they contain the same letters, regardless of order. We also use the following result which is a classical theorem of Dedekind. The proof of this, which depends on the axiom of choice, can be found on p.880 of [18]. Theorem 2.28. If G is a free abelian group then any subgroup of G is itself free abelian.

Sketch Proof of Proposition 2.24. As G is a ﬁnite group, there exists a free group F and surjective homomorphism π : F → G. We deﬁne

F φ : F → G × F 0 x 7→ (π(x), xF 0).

13 It is easy to check that φ is a homomorphism of groups. Now let T = φ(F ), then it is clear that

Z(T ) = φ π−1(Z(G)) . From Corollary 2.21, and the fact that φ is a homomorphism,

T φ(F ) F/ ker(φ) F = =∼ =∼ . T 0 (φ(F ))0 (F/ ker(φ))0 F 0 ker(φ) However, ker(φ) = ker(π) ∩ F 0 ⊂ F 0 and so T/T 0 =∼ F/F 0. Thus T/T 0 is a free abelian group. Moreover, Z(T )/(Z(T ) ∩ T 0) is isomorphic to Z(T )T 0/T 0 so is isomorphic to a subgroup of a free abelian group and is thus free abelian itself. As free abelian groups are simply the direct sums of many copies of Z, this implies that we can ﬁnd a (free abelian) group B such that

Z(T ) = B × (Z(T ) ∩ T 0).

T As B is a subgroup of Z(T ), it is normal in T . Let G1 = B and deﬁne two maps:

G G τ : → 1 Z(G) Z(G1)

π(f)Z(G) 7→ (φ(f)B)Z(G1)

for f ∈ F and

0 0 σ : G → G1 x 7→ (x, 1)B.

The rest of the proof, which we omit, then consists of showing that τ and σ are well-deﬁned group isomorphisms and that G1 is a ﬁnite group such that 0 Z(G1) ⊂ G1. These results follow almost directly from their constructions.

14 Chapter 3

Key Results on the Commuting Probability

3.1 Basic Results

Recall that we deﬁne the commuting probability on a ﬁnite group G as

|{(x, y) ∈ G × G | xy = yx}| Pr(G) = . |G|2 It is clear that Pr(G) = 1 if and only if G is abelian and also that, as the identity commutes with all elements, Pr(G) > 0 for all G ∈ G. However, the link between the commuting probability of a group and its structure extends much further than this. We first note that P |CG(x)| Pr(G) = x∈G . |G|2 A fact which follows directly from the definition. We then use this to show one of the most fundamental properties of this probability: Theorem 3.1 (Gustafson [10]). If G is a finite group,

k Pr(G) = |G|

where k = k(G) is the number of conjugacy classes of G.

Proof. Let C = {(x, y) ∈ G × G | xy = yx} then, as above, we have X |C| = |CG(x)|. x∈G

15 −1 −1 It is easy to show that if x = g yg, then CG(x) = g CG(y)g. Furthermore, the number of elements in the conjugacy class of x is [G : CG(x)]. Thus, if x1, . . . , xk are representatives of the conjugacy classes in G we have

k X |C| = [G : CG(xi)] · |CG(xi)| = k · |G| i=1 k and Pr(G) = |G| . Another basic result is that the probability is multiplicative on direct prod- ucts:

Theorem 3.2. For G and H ﬁnite groups,

Pr(G × H) = Pr(G) · Pr(H).

Proof. From above, k(G) Pr(G) = . |G| We claim that k(G × H) = k(G)k(H). Let n = k(G) and m = k(H). Then take gi, . . . , gn and h1, . . . , hm to be representatives of the conjugacy classes of G and H respectively. Then, for 1 ≤ i ≤ n, 1 ≤ j ≤ m,

−1 −1 ClG(gi) × ClH (hj) = {(ggig , hhjh ): g ∈ G, h ∈ H} −1 = {(g, h)(gi, hj)(g, h) :(g, h) ∈ G × H}

= ClG×H ((gi, hj)).

Thus (gi, hj) are representatives of the conjugacy classes in G × H for 1 ≤ i ≤ n, 1 ≤ j ≤ m and k(G × H) = mn, as required. Also of note is the relationship between the commuting probability of a group and that of its subgroups. Proposition 3.3. If H ≤ G then

Pr(G) ≤ Pr(H).

Proof. For x ∈ G, if we let CH (x) = H ∩ CG(x) then CH (x) ≤ CG(x). Suppose that g1 and g2 are representatives of two distinct cosets of CH (x) in CG(x). We claim that g1H and g2H are distinct cosets of H in G. If this is not the case then g1 = g2h for some h ∈ H. Then

−1 h = g2 g1 ∈ CG(x) ∩ H = CH (x) so g1CH (x) = g2CH (x), a contradiction. Thus we have

16 [CG(x): CH (x)] ≤ [G : H]. (3.1) −1 If x ∈ G and y ∈ CH (x) then yxy = x so y ∈ H and x ∈ CG(y). Thus X X |CH (x)| = |CG(y)|. x∈G y∈H Combining this with the inequality in (3.1) gives

X 2 X |CG(x)| ≤ [G : H] |CH (y)|. x∈G y∈H Dividing through by |G|2, we get Pr(G) ≤ Pr(H).

Proposition 3.4 ([7]). If N E G then Pr(G) ≤ Pr(G/N)Pr(N)

Proof. As N E G, for x ∈ G, CN (x) = N ∩ CG(x) E CG(x). By the second isomorphism theorem,

CG(x) ∼ CG(x)N = ⊆ CG/N (xN). CN (x) N So |CG(x)| ≤ |CG/N (xN)| · |CN (x)|. Summing over G gives X X |CG(x)| ≤ |CG/N (xN)| · |CN (x)| x∈G x∈G X X = |CG/N (xN)| |CN (y)|. xn∈G/N y∈xN

Now let z ∈ N and suppose that CxN (z) = CG(z) ∩ xN 6= φ for some x ∈ G. −1 Then there exists a ∈ CxN (z), say a = xn for some n ∈ N. Then xnz(xn) = z −1 −1 −1 and nzn = x zx. So nzn ∈ ClG(z) so n is in a coset of z in N. Then CxN (z) is a coset of Cn(z) and

( |CN (z)| if CxN (z) 6= φ |CxN (z)| = 0 if CxN (z) = φ.

17 In particular, |CxN (z)| ≤ |CN (z)| and X X X |CG(x)| ≤ |CG/N (xN)| |CN (z)| x∈G xN∈G/N z∈N as required. An important area of the study of the commuting probability has been on bounding it for certain types of groups. We prove below the most basic of these results before giving a full overview of the topic in Section 4.1. This result appears to have ﬁrst been noted by Gustafson [10]:

Proposition 3.5. If G is not abelian, then

Pr(G) ≤ 5/8.

Proof. If C1,...,Ct are the non-trivial conjugacy classes of G then

|G| = |Z(G)| + |C1| + ... + |Ct| ≥ |Z(G)| + 2t.

So k = t+|Z(G)| ≥ (|G|+|Z(G)|)/2. As G is non-abelian, G/Z(G) is not cyclic so |Z(G)| ≤ |G|/4. Thus k ≤ 5/8 · |G| so Pr(G) ≤ 5/8.

Moreover, this bound is attained by, for example, Q8. The group repre- 2 2 2 2 sentation of Q8 is h−1, i, j, k | (−1) = 1, i = j = k = ijk = −1i. So the elements 1 and −1 commute with all elements of the group. If x is any of the other elements then it commutes with 1, −1, x and −x and 2 · 8 + 6 · 4 5 Pr(Q ) = = . 8 8 · 8 8 3.2 The Commuting Probability and Character Degrees

Here, we recall and prove Theorem 1.2. Theorem 3.6 (Guralnick and Robinson [9]). If G is non-abelian, we have

1 1 1 Pr(G) ≤ + 1 − . d2 d2 |G0| Where d is the smallest degree of a non-linear complex irreducible character of G. In order to prove this, we ﬁrst recall two important facts regarding the characters of ﬁnite groups:

• The number of irreducible characters of G is equal to k(G), the number of conjugacy classes.

18 • If x1, . . . , xk are the degrees of the irreducible characters of G then 2 2 x1 + ... + xk = |G|.

We must also prove the Lemma: Lemma 3.7. If G is a ﬁnite group, the number of linear irreducible complex characters of G is equal to [G : G0]. Proof. Let θ be the projective map G → G/G0. If χ : G/G0 → C× is a linear character of G/G0 then χ ◦ θ is a linear character of G. In fact, we will show that every linear character of G is of this form. Ifχ ˆ : G → C× is a linear character then it is a homomorphism onto an abelian group so for all x, y ∈ G,

χˆ(x−1y−1xy) =χ ˆ(x)−1χˆ(y)−1χˆ(x)ˆχ(y) = 1 so G0 ⊆ kerχ ˆ. Thus, if we let χ : G/G0 → C× be deﬁned as χ(θ(x)) =χ ˆ(x), it is well deﬁned as

gG0 = hG0 ⇒ g−1h ∈ G0 ⇒ χˆ(g−1h) = 1 ⇒ χˆ(g) =χ ˆ(h). Thusχ ˆ is of the form χ◦θ and we have a one-to-one correspondence between the linear characters of G and the linear characters of G/G0. Finally, as G/G0 is abelian, all characters are linear and their number is equal to the number of conjugacy classes which, as G/G0 is abelian, is equal to |G/G0|. Proof of Theorem 1.2. Let k = k(G) and m = [G : G0] then

|G| = m + x2 + ... + x2 . n1 nk−m

Where xni are the degrees of the non-linear irreducible characters of G. If d = min1≤i≤k−m(xni ) then

|G| ≥ m + (k − m)d2 = m(1 − d2) + kd2 which implies

|G| 1 k ≤ + m 1 − . d2 d2 Finally, dividing through by |G| gives

1 1 1 Pr(G) ≤ + 1 − d2 d2 |G0| as required.

19 3.3 Commuting Probability of Semigroups

The commuting probability of semigroups is conceptually very similar to that of groups but the two actually behave in very different ways. It is their differences, rather than their similarities which make this topic interesting to discuss. Definition 3.8. A semigroup is a set together with a binary operation · such that ∀x, y, z ∈ S: • x · y ∈ S. • (x · y) · z = x · (y · z) (associativity). A semigroup may be thought of as a generalisation of a group; a group is a semigroup with an identity element such that every element has an inverse. As with groups, we usually write xy in place of x · y. Unsurprisingly, we define the commuting probability of the semigroup to be

|{(x, y) ∈ S × S : xy = yx}| Pr(S) = . |S|2 We have already seen that there are large “gaps” in the range of commuting probability of groups. For example, Proposition 3.5 implies there are no finite groups G such that Pr(G) ∈ (5/8, 1). In the case of semigroups however, the picture is completely different. In particular, in 2008, Berta Givens [8] showed that the set of possible commuting probabilities for semigroups is dense in (0, 1]. In 2012, Ponomarenko and Selinki [22] improved this result. They showed by construction of families of semigroups that Theorem 3.9. For every rational number x ∈ (0, 1], there exists a semigroup S such that Pr(S) = x. Their result relies heavily on the famous result of Lagrange (see, for example, pp. 281-2 [13]): Theorem 3.10. Every natural number can be expressed as the sum of four integer squares. In particular, unless the number is of the form 4k(8m + 7), it can be expressed as the sum of three squares. They use this to identify four families of semigroups whose probabilities are exactly Q ∩ (0, 1]. To illustrate their method, we will prove the result for the 1 rational numbers in (0, 3 ]. The remaining cases follow in a similar fashion. Lemma 3.11. For every rational number x ∈ (0, 1/3], there exists a semigroup S such that Pr(S) = x. Proof. We aim to construct a family, S(a, b, c, k), of semigroups whose com- 1 muting probabilities are precisely the rational numbers in (0, 3 ]. Take the set S = A∪B ∪C ∪D1 ∪...∪Dk comprised of sets such that |A| = a, |B| = b, |C| = c, |D1| = ... = |Dk| = 2. Let α ∈ A, β ∈ B, γ ∈ C, δ1 ∈ D1, . . . , δk ∈ Dk. Now define the map f : S → S as

20 α if x ∈ A,  β if x ∈ B, f : x 7→ γ if x ∈ C,  δi if x ∈ Di.

Then deﬁne the semigroup operation as x · y = f(x). Then the commuting probability is

a2 + b2 + c2 + 4k . (a + b + c + 2k)2 Now let p/q ∈ (0, 1/3] and deﬁne M = 16pq − 8q + 3. As this is not in the form 8k(8m + 7), there exists integers x, y, z such that M = x2 + y2 + z2. If we set 4q − a − b − c a = x + 1, b = y + 1, c = z + 1, k = 2 then we ﬁnd that the commuting probability of S(a, b, c, d) is p/q.

Recently, this result has been further reﬁned. Soule [26] showed that it is possible to prove Theorem 3.10 using only one family of semigroups. For n ∈ N and X = {x1, x2, . . . , xm} ⊆ {1, . . . , n} with m ≤ n and x1 < x2 < . . . < xm, she deﬁned S(n, X) to be the semigroup with underlying set {1, . . . , n} and multiplication

( x if x < a ≤ x and b ≤ x a · b = i i−1 i m max{a, b} if a > xm or b > xm.

She then procedes to show that this does indeed form a semigroup, and that the commuting probabilities of semigroups in this family consists of all rational numbers in (0, 1], as required.

21 Chapter 4

Background

In this Chapter, we outline the most important results in the commuting probability over the past century, as well as giving details of results which we later use in our own method. By Theorem 3.1, studying the commuting probability of a group is equivalent to studying the number of conjugacy classes it has. In fact, many of the early results in this ﬁeld are presented from this perspective.

4.1 Bounds on The Commuting Probability

If G is a ﬁnite group with k conjugacy classes and order n then trivially k ≤ n. The ﬁrst person to consider an upper bound for n in terms of k was Landau [17]. He made the basic observation that, if C1,...,Ck are the conjugacy classes and hi = |Ci| for 1 ≤ i ≤ k then

−1 −1 (n/h1) + ... + (n/hk) = 1. Using this, he showed that such a bound exists, although did not ﬁnd it explicitly. The ﬁrst to do so was Brauer [2] in 1963. He continued Landau’s work in his essay Representations of of Finite Groups and used Landau’s method to show that for k ≥ 3,

k−1 k−3 Y i−2 n ≤ (2k)2 i2 . i=1 In particular, this means that as n grows larger, k must too. Notably, he remarks that, since Landau’s method used very little group theory, better bounds may well be possible and suggests this as an open problem. In 1968, Erd˝osand Tur´an[6] and, independently, Newman [21], found such a bound:

Theorem 4.1. For any ﬁnite group G,

22 log log |G| Pr(G) ≥ 2 2 . |G| We will outline Erd˝osand Tur´an’sproof, although that of Newman is very similar; both are based on Landau’s paper. They use the following lemma, the proof of which may be found in Curtiss [4]:

Lemma 4.2. For v ∈ N ﬁxed, let x1, x2, . . . , xv be positive integers such that 1 1 1 + + ... + < 1. x1 x2 xv Then 1 1 1 1 + + ... + < 1 − x1 x2 xv αv+1 − 1

where αv is deﬁned recursively as

α1 = 2, αk+1 = α1α2 . . . αk + 1.

Proof of Theorem 4.1. Again take n to be the order of the group and h1, h2, . . . , hk to be the sizes of its conjugacy classes (where k = k(G)). Then setting xi = n/hi gives 1 1 1 + + ... + = 1 x1 x2 xk As there is a conjugacy class of size 1 (that which contains the identity), we can assume that hk = 1. Then xk = n and 1 1 1 1 + + ... + = 1 − < 1 x1 x2 xk−1 n so we may apply the Lemma with v = k − 1. This gives 1 1 1 − < 1 − n αk − 1 2 2 i.e. αk > n + 1. From the deﬁnition of the αk, αv+1 = αv − αv + 1 < αv so 2k 2k αk < 2 . Then 2 > n and so

k > log2 log2 n as required. The topic was then left largely untouched until 1992 when Pyber [23] proved that

23 Theorem 4.3. There exists an explicitly computable constant > 0 such that every group of order n ≥ 4 satisﬁes

log2 n Pr(G) ≥ 8 . n(log2 log2 n) This is considered to be an answer to Brauer’s question on the bound of the number of conjugacy classes. In recent years, this result has been improved by Keller [16]: Theorem 4.4. There exists an explicitly computable constant > 0 such that every group of order n ≥ 4 satisﬁes

log2 n Pr(G) ≥ 7 . n(log2 log2 n) This is currently the best known lower bound for the number of conjugacy classes of a ﬁnite group in terms of the order of the group.

4.2 Classiﬁcation Results

Some recent work has taken a slightly diﬀerent approach, classifying groups with Pr(G) above a certain value. This is an area which has recieved slightly less attention but the methods used in the following papers are of particular note to our problem.

4.2.1 Classification of Groups with Pr(G) > 11/32 The first, and most notable, work in this area was done by Rusin in 1979. He classifies all finite groups for which Pr(G) > 11/32. Of particular importance both to his method and to ours is a result he proves giving an explicit formula for the commuting probability of p-groups. Before stating it, we first introduce some notation. For H ⊆ G,

H∗ := {x ∈ G :[G, x] ⊆ H}.

This operator can be thought of as a partial inverse to ( )0 as

(H∗)0 ⊆ H and (H0)∗ ⊆ H.

Theorem 4.5. If G is a p-group with G0 ≤ Z(G), then ! 1 X (p − 1)[G0 : K] Pr(G) = 1 + (4.1) |G0| pn(K)+1 K where the sum is taken over all strict subgroups K of G0 such that G0/K is cyclic and the integer n(K) is deﬁned as

pn(K) := [G : K∗].

24 The proof of this result relies on the idea of Mobiüsinversion. The following is a standard result which may be found in Cameron [3], for example. Proposition 4.6. Suppose that X is a partially ordered set (poset) and that f and g map X to an abelian group. If we define the Möbiusfunction on X by X m(A, A) = 1 and m(A, C) = − m(A, C) ∀A, C ∈ X A≤B

! 1 X |H¯ | 1 X 1 X Pr(G) = = m(K,H)|K∗| |G| |H| |G| |H| H≤G0 H≤G0 K

The M¨obiusfunctions for the subgroups of p-groups has been calculated by Weisner [27]. If K is not normal in H, m(K,H) = 0; otherwise m(K,H) = m(1,H0) where H0 = H/K. Then   1 X X m(1,H0) Pr(H) = |K∗| . |G|  |K| · |H0| K≤G0 H0≤G0/K Weisner also calculated m(1,H0). If H0 is an elementary abelian p-group of order pi, m(1,H0) = (−1)ipi(i−2)/2; otherwise it is zero. Then   1 X |K∗| X m(1,H0) Pr(H) = |G| |K|  |H0|  K≤G0 H0≤L

25 where L is the subgroup of elements of order p or less in G0/K. Now L must be isomorphic to a vector space of dimension n for some n ∈ N. If α(n, j) denotes the number of subgroups of order pj (equivalently subspaces of dimension pj) in L then

α(n, j) = pjα(n − 1, j) + α(n − 1, j − 1) and α(n, 0) = α(n, n) = 1.

Using this, we can show that

0 n i n X m(1,H ) X m(1, (Cp) )α(n, i) X = = (−1)ipi(i−3)/2α(n, i) |H0| pi H0≤L i=0 i=0  1 if n = 0,  = 1 − (1/p) if n = 1,  P (1 − (1/p)) n−1 m(1,H) if n ≥ 2. H≤(Cp) To evaluate the last sum, we again appeal to M¨obiusinversion. Deﬁne a function n−1 on the subgroups of (Cp) as f({e}) = 1 and f(H) = 0 for H 6= {e}. Then P P let g(H) = K≤H f(K) ≡ 1 giving f(H) = K≤H m(K,H)g(K). As n ≥ 2, n−1 (Cp) 6= {e} so

n−1 0 = f((Cp) ) X n−1 = m(K,Cp )g(K) n−1 K≤(Cp) X n−1 = m(1,Cp /K) n−1 K≤(Cp) X = m(1,H). n−1 H≤(Cp)

We can now evaluate Pr(G). If n = 0, L = {e} and G0/K has no elements of order p, so K = G0. If n = 1, G0/K has a unique subgroup of order p and, as it is already abelian, this is when G0/K is cyclic (and non-trivial). If n ≥ 2, as we have just seen, K will not contribute to the sum. So  1 if K = G0 1 X |K∗|  Pr(G) = · 1 − (1/p) if G0/K cyclic |G| |K| K≤G0 0 otherwise as required. Another significant result he uses concerns groups where G0 ∩ Z(G) = {e}: Proposition 4.7 (Rusin, 1979). For any fixed finite group, G, there exists at most a finite number of groups K such that G0 =∼ K0 and Z(K) = {e}. He uses the following result, known informally as the ‘N over C Theorem’:

26 Theorem 4.8. If H ≤ G, NG(H)/CG(H) is isomorphic to a subgroup of Aut(H).

Proof. Let f : NG(H) → Aut(H) map g to conjugation by g. Then f is a homomorphism with ker(f) = CG(H) and the claim follows. We also recall that the rank of a group G is deﬁned as

rank(G) = min{|X| | X ⊆ G, such that G = hXi}. Proof of Proposition 4.7. Our aim is to bound the size of K in terms of the order of G. From Theorem 4.8,

0 K NK (K ) L = 0 = 0 CK (K ) CK (K )

is isomorphic to a subgroup of Aut(K0). Now, by Corollary 2.21,

0 0 0 K CK (K ) L = 0 CK (K ) and

0 L K/CK (K ) ∼ K 0 = 0 0 0 = 0 0 . L (K CK (K ))/CK (K ) K C(K ) 0 0 0 which is, of course, abelian. Let rank(L/L ) = n, then K/(K CK (K )) can be generated by n elements xi for 1 ≤ i ≤ n. 0 0 0 From Lemma 2.3, (CK (K )) ≤ Z(K) = {e} which implies that CK (K ) 0 0 0 is abelian. Furthermore, if y ∈ CK (K ), then [K CK (K ), y] = {e}. Since 0 0 0 K = hx1, . . . , xn,K C(K )i then if y ∈ CK (K ) commutes with xi for each 1 ≤ i ≤ n then y ∈ Z(G) = {e}. 0 Therefore, if for some y1, y2 ∈ CK (K ), we have [y1xi] = [y2xi] for all i, then −1 −1 0 y1y2 commutes with xi so y1y2 = e and y1 = y2. In particular, |CK (K )| is bounded above by the number of values the n-tuple {[y, xi] | 1 ≤ i ≤ n} may 0 take as y ranges over CK (K ). Thus

n n Y 0 Y 0 n |[CK (K ), xi]| ≤ |[K, xi]| ≤ |K | . i=1 i=1 0 0 Then, as |K| = |CK (K )| · |K/CK (K )|,

0 |K| ≤ |K0|n · |L| ≤ |K0||Aut(K )||Aut(K0)|. As K0 =∼ G0, the order of K is bounded by a function of G and so the possibilities it may take is ﬁnite.

27 We now state Rusin’s main theorem, as well as outlining the key steps in its proof: Theorem 4.9. If Pr(G) > 11/32 then G satisﬁes one of the following possibilities: Pr(G) G0 G0 ∩ Z(G) G/Z(G) −2s 2s (1 + 2 )/2 C2 C2 (C2) 1/2 C3 {e} S3 7/16 C4 or C2 × C2 C2 D4 3 4 C2 × C2 C2 × C2 C2 orC2 11/27 C3 C3 C3 × C3 2/5 C5 {e} D5 3 4 25/64 C2 × C2 C2 × C2 C2 or C2 3/8 C6 C2 C2 × S3 or T

Where T is the non-abelian group of order 12 which is not isomorphic to A4 or C2 × S3. Sketch Proof of Theorem 4.9. From Theorem 1.2, if Pr(G) > 11/32, |G0| ≤ 7. Rusin’s technique is to consider groups G such that |G0| = k for 1 ≤ k ≤ 7, finding all such groups with Pr(G) > 11/32. If G0 = {e}, G is abelian so trivially satisfies Pr(G) > 11/32. For remaining cases, he simplifies them further, splitting them into subcases, depending on G0 ∩ Z(G). However, some possibilities for G0 and G0 ∩ Z(G) are 0 ∼ not possible. From Section 2.2.3, there are no groups for which G = S3 and if 0 ∼ 0 G = C2k then G ∩ Z(G) must be even. Thus the cases which remain to be checked are:

0 G C2 C3 C4 C2 × C2 C5 C6 C7 0 G ∩ Z(G) C2 {e} C2 {e}{e} C2 {e} C3 C4 C2 C5 C6 C7 C2 × C2

Rusin considers two principal possibilities:

• G0 < Z(G) • G0 ∩ Z(G) = {e}.

There are of course two cases which do not ﬁt into either of these categories; 0 0 0 ∼ 0 ∼ |G | = 4 and |G ∩ Z(G)| = 2, and G = C6 and G ∩ Z(G) = C2. Both these cases Rusin deals with separately by explicitly calculating their commuting probabilities. If G0 < Z(G), then G is nilpotent of class 2 so is a product of its Sylow 0 0 0 subgroups. In particular, if G = H1 × ... × Hk then G = H1 × ... × Hk and Pr(G) = Pr(H1) × ... × Pr(Hk). So we need only consider non-abelian Sylow subgroups, of which there are a limited number.

28 0 ∼ If G = Cp then G has one non-abelian p-Sylow with commutator subgroup 0 0 ∼ G . If G = C4 or C2 × C2 then G has one non-abelian 2-Sylow with com- 0 0 ∼ mutator subgroup G . If G = C6 then G has one non-abelian 2-Sylow and one non-abelian 3-Sylow, with commutator subgroups isomorphic to C2 and C3 respectively. In each of these cases, Rusin directly applies Theorem 4.5 to calculate Pr(G) and classify the cases where it is greater than 11/32. In the case where G0 ∩ Z(G) = {e}, he shows that Pr(G) = Pr(G/Z(G)). Moreover, from Corollary 2.21,

G 0 G0Z(G) G0 =∼ =∼ = G0 Z(G) Z(G) G0 ∩ Z(G) and G (G0 ∩ Z(G))∗ {e}∗ Z(G) Z = =∼ = = {e}. Z(G) Z(G) Z(G) Z(G) So Pr(G) = Pr(K) for some group K such that G0 =∼ K0 and Z(K) = {e}, considerably reducing the problem. 0 ∼ He also show that, if K = Cp for p prime and Z(K) = {e}, then K = ha, b | ap = bn = e, bab−1 = ari, where n|p − 1 and rj ≡ 1 mod p if and only if n|j. In this case we have

n2 + p − 1 Pr(G) = . n2p He then uses this result to derive explicit formulae for the commuting probability of groups in this case, and classify the groups for which Pr > 11/32.

4.2.2 Classiﬁcation of Groups of Odd Order with Pr(G) > 11/75 The work of Rusin was extended in 2011 by Das and Nath [5] who classify groups G of odd order such that Pr(G) > 11/75. Their most signiﬁcant result is the following Theorem which gives the commuting probability for groups G such that |G0| = p2 and |G0 ∩Z(G)| = p where p is a prime such that gcd(p−1, 1) = 1. Theorem 4.10. Let p be a prime such that gcd(p − 1, 1) = 1. If |G0| = p2 and |G0 ∩ Z(G)| = p then there exists s ∈ N such that

 2 2p −1 0  4 if CG(G ) is abelian , Pr(G) = p 1 p−1 2  p4 p2s−1 + p + p − 1 otherwise.

The result relies on few preliminary results:

0 0 Lemma 4.11. For a group G, G/CG(G ) can be embedded in Aut(G ), and so, 0 0 |ClG(x)| divides |Aut(G )| for all x ∈ G .

29 Proof. The ﬁrst claim follows from Theorem 4.8, and the fact that G0 is normal 0 so NG(G ) = G. The second claim follows from the ﬁrst and the fact that [G : CG(x)] = |Cl(x)| for all x ∈ G. Proposition 4.12. Let p be a prime such that gcd(p−1, |G|) = 1. If |G0|/|G0 ∩ Z(G)| = p, then G is nilpotent of class 3. Proof. If H = G/Z(G) then from Corollary 2.21

G G Z(G) {e} = =∼ 2 . Z(G) 2 Z(G)

So G2Z(G) = Z(G) and G2 ≤ Z(G). Thus G is nilpotent of class at most 3, as required.

Lemma 4.13. If G0 6⊆ Z(G) and p is a prime such that one of the following conditions hold:

0 ∼ • G = Cp2 and gcd(p − 1, |G|) = 1 0 ∼ 2 • G = Cp × Cp and gcd(p − 1, |G|) = 1

G 0 then | 0 | = |ClG(x)| = p for all x ∈ G − Z(G). CG(G )

G ∼ 0 Lemma 4.14. If 0 = Cp for p a prime, then |ClG(x)| divides |G | for all CG(G ) 0 x ∈ G − CG(G ). G ∼ 0 0 Proof. As 0 = Cp, if x ∈ G − CG(G ) then hCG(G ), xi = G and so CG(G ) 0 G 0 0 [G, x] = [CG(G ), x]. As 0 is abelian, [CG(G ), x] is a subgroup of G , and CG(G ) the result follows from the fact that |[G, x]| = |ClG(x)|. Lemma 4.15. If p is the smallest prime divisor of |G| and |Cl(x)| = p then 0 x ∈ CG(G ).

Proof. If x ∈ G such that |ClG(x)| = p, then G/CG(x) forms an abelian group 0 0 so G ⊆ CG(x) which implies x ∈ CG(G ).

30 Lemma 4.16 ([1], p. 303). If G is non-abelian, H E G and [G : H] = p, where p is a prime, then

Pr(H) p + 1 X Pr(G) = + |C (x)|. p2 p|G|2 G x∈G−H

Proof of Theorem 4.10. From Proposition 4.12, G is nilpotent. In particular, it is the product of its Sylow subgroups. Moreover, if H1,...,Hk are its Sylow 0 0 0 0 ∼ subgroups then G = H1 × ... × Hk but G = Cp so we must have G = H × A where H is a p-group and A is abelian. Thus, we may assume that G is a p-group. Then we have gcd(p − 1, |G|) = 1 and gcd(p2 − 1, |G|) = 1 so by Lemma 0 0 4.13, [G, : CG(G )] = p. From Lemmas 4.14 and 4.15, if x ∈ G − CG(G ) then 2 0 |ClG(x)| = p . Then setting H = CG(G ) in Lemma 4.16 gives:

Pr(C (G0)) p2 − 1 Pr(G) = G + . p2 p4

0 If CG(G ) is abelian, the result follows directly. Otherwise, from Lemma 2.3 0 0 and the fact that |G ∩ Z(G)| = p, CG(G ) is nilpotent and is equal to K × B where K is a p-group and B is abelian. Thus from Theorem 4.5

1 p − 1 Pr(C (G0) = 1 + G p p2s 2s 0 0 where p := [CG(G ): Z(CG(G ))], giving the desired result. Theorem 4.17. If |G| is odd and Pr(G) ≥ 11/75, then G satisﬁes one of the following possibilities:

Pr(G) G0 G0 ∩ Z(G) 1 {e}{e} 2s (1 + 2/3 )/3 C3 C3 2s (1 + 4/5 )/5 C5 C5 5/21 C7 {e} 55/343 C7 C7 17/81 C9 or C3 × C3 C3 C3 × C3 C3 × C3 121/729 C3 × C3 C3 × C3 7/39 C13 {e} 3/19 C15 {e} 29/189 C21 C3 11/75 C5 × C5 {e} Sketch proof of Theorem 4.17. From Theorem 1.2, if Pr > 11/75 and |G0| is odd then |G0| ≤ 25. Like Rusin, they split each of these possibilities into subcases depending on G0 ∩Z(G). From [20], G0 cannot be isomorphic to the non-abelian

31 0 ∼ 0 group of order 21. Das and Nath also show that if G = C15 then G ⊆ Z(G). Thus the cases to be checked are:

0 G C3 × C3 C9 C15 C21 C5 × C5 C25 Cp 0 G ∩ Z(G) {e}{e} C15 {e}{e}{e}{e} C3 C3 C3 C5 C5 Cp C3 × C3 C9 C7 C5 × C5 C25 C21 Like Rusin, Das and Nath then split the problem down into two principal cases:

• G0 ⊆ Z(G) • G0 ∩ Z(G) = {e}

If G0 ⊆ Z(G), then they appeal directly to Theorem 4.5, Rusin’s result on the commuting probability of p-groups. From this they calculate explicitly the possible values of Pr(G) and determine the cases where this is greater than or equal to 11/75. 0 0 If G ∩ Z(G) = {e}, by Lemma 2.3, CG(G ) is abelian. They show that, in this case, G0 may not have order 3,5,9,17 or 21. If G0 is of prime order, other than these cases, they use the result of Rusin which says that P r(G) = (n2 + p − 1)/(pn2) where n is a divisor of p − 1. From this they calculate explicitly the possible values of Pr(G) and classify the groups for which this is above 11/75. The only remaining possibility is |G0| = 25. In this case, they 0 ∼ show that G = C5 × C5 and that Pr(G) = 11/75. There are a few cases which do not ﬁt into either of these categories. In the cases |G0| = 9, |G0 ∩ Z(G)| = 3 and |G0| = 25, |G0 ∩ Z(G)| = 5 , they use 0 ∼ Theorem 4.10 to calculate Pr(G). The ﬁnal case is G = C21. They show that, 0 ∼ in this case, G ∩ Z(G) = C3 and that 17/81 is the only value of Pr(G) which lies in [11/75, 1].

32 Chapter 5

Joseph’s Conjectures

Recall that in 1977 Joseph [15] conjectured that:

1. Every limit point of Pr(G) is rational.

2. If l is a limit point of Pr(G), then there exists = l > 0 such that Pr(G) ∩ (l − l, l) = φ.

3. Pr(G) ∪ {0} is a closed subset of R. Where G is the set of all finite groups. Recall also that Hegarty [12] has proved that Theorem 5.1. The set Pr(G) ∩ (2/9, 1] is good (i.e. it satisfies Joseph’s first two conjectures). In this section we will outline the proof of this result as well as suggesting ways in which to improve the bound.

5.1 Hegarty’s Work

As we will see in Section 5.1.2, the assumption that Pr(G) > 2/9 is used on a number of occasions in Hegarty’s proof. However, the only instance where the bound must be exact (i.e a lower bound would not suﬃce) is in showing that Pr(G) > 2/9 implies that either |G0| < 8 or that d = 2, where d is the smallest degree of a non-linear irreducible character of G. From the discussion following Theorem 1.2 in the introduction, we see that one way to improve Hegarty’s result would be to prove that Joseph’s conjectures hold for groups such that |G0| = 8. We discuss this in Section 5.2.

5.1.1 Groups with |G0| < 8 In this section we provide a full proof of the claim that the set of groups with |G0| < 8 form a good set.

33 Lemma 5.2. Let C := {G ∈ G : |G0| = 8}. Then Pr(C) is a good set. Proof. If |G0| = 1 then G is abelian so Pr(G) = 1 and we are finished. The remaining cases are treated one-by-one. By the results on isoclinic groups in Propositions 2.24 and 2.23, for any finite group G, there exists a finite group H such that Pr(G) = Pr(H) and Z(H) ≤ H0 so we may assume that Z(G) ≤ G0. 0 ∼ Moreover, from Section 2.2.3, there are no groups for which G = S3 and if 0 ∼ G = C2k then Z(G) must be even. Thus the cases which we must check are:

0 2 G C2 C3 C4 C2 C5 C6 C7 Z(G) C2 {e} C2 {e}{e} C2 {e} C3 C4 C2 C5 C6 C7 2 C2

Groups with Z(G) = {e}. In those cases where it is possible to have Z(G) = {e}, Rusin’s result Propostion 4.7 implies that G may be isomorphic to only a ﬁnite number of groups. In particular, Pr(G) may take only a ﬁnite number of values so will satisfy Joseph’s conjectures.

Groups with G0 = Z(G). If G0 = Z(G) then G is nilpotent and is thus the product of its Sylow subgroups. If G = H1 × ... × Hk where Hi are the Sylow 0 0 0 subgroups, then G = H1 × ... × Hk and Pr(G) = Pr(H1) × ... × Pr(Hk). Thus, we need only consider the non-abelian Sylow subgroups of G. We then use Theorem 4.5 which says that if H is a p-group then: ! 1 X (p − 1)[H0 : K] Pr(H) = 1 + (5.1) |H0| pn(K)+1 K where the sum is taken over all strict subgroups K of H0 such that H0/K is cyclic and the integer n(K) is deﬁned as

|H| pn(K) := . |K∗| 0 ∼ If G = Cp for p prime then G has one non-abelian Sylow subgroup, namely a p-Sylow with commutator subgroup Cp so 1 p − 1 Pr(G) = 1 + p p2s where p2s := [G0 : Z(G)]. This is clearly of the form in Lemma 1.1 If G0 is a non-cylic p-group then G will again have one non-abelian Sylow subgroup, a p-Sylow with commutator subgroup G0. Then we will again use Rusin’s result (5.1). As K is a subgroup of G0,[G0 : K] = pb where b ≤ s(K)+1 so Pr(G) will be of the form in Lemma 1.1.

34 0 ∼ Finally, if G = C6 then G has two non-abelian Sylow subgroups, a 2-Sylow, H1 with commutator subgroup C2 and a 3-Sylow, H2 with commutator subgroup C3. Then, by Rusin’s equation, (5.1), 1 1 1 2 Pr(H1) = 1 + and Pr(H2) = 1 + . 2 22s1 3 32s2 So 1 1 2 2 Pr(G) = 1 + + + . 6 22s1 32s2 22s1 32s2 Which is again of the form we require.

Groups with |G0| = 4, |Z(G)| = 2. In this case, Das and Nath [5] have calculated that

1 1 1 Pr(G) = 1 + + 4 4 22s+1 2s 0 0 where 2 := [CG(G ): Z(CG(G ))].

0 ∼ ∼ Groups with G = 6, Z(G) = C2. In this case, Rusin [25] has calculated that 0 ∼ 0 ∼ if G = C6 and G ∩ Z(G) = C2 then 1 1 Pr(G) = + 4 2s for some s ≥ 3. He provides no proof for this fact, cryptically stating that the calculations are “rather involved, and not particularly interesting”.

5.1.2 The Main Proof In this section we give an outline of Hegarty’s proof of Theorem 5.1. The proof relies heavily on the following result:

Lemma 5.3. Let n ∈ N and let (A)n be the collection of all finite groups possessing a normal abelian subgroup of index at most n. Then Pr(An) is a good set. Partial Proof of Theorem 5.1. From Theorem 1.2, as Pr(G) > 2/9, either |G0| < 8 or d = 2 where d is the smallest degree of a non-linear irreducible character of G. If |G0| < 8 then we showed in Section 5.1.1 that the conjectures are satisfied. So we may assume that G has an irreducible representation, φ of degree 2. Let π : GL(2, C) → P GL(2, C) be the natural projection and let L : ker(π ◦ φ). Then G/L is isomorphic to a finite subgroup of P GL(2, C). Thus G/L must be isomorphic to one of A4, S4, A5 or D2n for n ≥ 2. We can calculate that ( 1 5 1 n+6 if n is even, Pr(A ) = , Pr(S ) = , Pr(A ) = , Pr(D ) = 4n 4 3 4 24 5 12 2n n+3 4n if n is odd.

35 ∼ If G/L = S4 then from Proposition 3.4, Pr(G) ≤ Pr(G/L)Pr(L) ≤ 5/24 < 2/9 ∼ ∼ so G/L =6 S4. If G/L = A5 then Pr(G) ≤ Pr(G/L)Pr(L) ≤ 1/12 < 2/9 ∼ ∼ so G/L =6 A5. If G/L = A4 then Pr(L) ≥ 3 · Pr(G) = 2/3 > 5/8. ∼ So if G/L = A4 then L is a normal abelian subgroup of bounded index and ∼ n 2 2 the result follows from Lemma 5.3. So we may assume G/L = DnhLa, Lb | a , b , (ab) ∈ Li. The rest of the proof is split into three cases:

Case 1: n ≥ 15.

If n ≥ 15, 4n 60 2 40 Pr(L) ≥ Pr(G) ≥ · = > 5/8. n + 6 21 9 63 We can then show that, as Pr(G) > 0, there exists some k = O(1) independent of n such that hL, aki is normal of bounded index and we may again appeal to Lemma 5.3.

Case 2: 3 ≤ n ≤ 14.

In this case, |G/L| is bounded. If L were abelian then [L : Z(L)] and, by extension [G : Z(L)], would be bounded so we would be able to use Lemma 5.3 with Z(L). Thus we may suppose that L is not abelian. As Pr(G/L) ≤ 5/8, 8 2 16 11 Pr(L) ≥ · = ≥ . 5 9 45 32 So by Rusin’s classiﬁcation, either [L : Z(L)] is bounded or |L0| = 2 and ∼ 2s L/Z(L) = C2 . In the ﬁrst case, Z(L) would again satisfy the conditions of Lemma 5.3. In the second, we can show that either |G0| ≤ 6, which we treat in Section 5.2 or that there exists a normal abelian subgroup of bounded index, and that we can use Lemma 5.3.

Case 3: n = 2.

In this case, Hegarty follows similar reasoning to the case 3 ≤ n ≤ 14 to show that either G has a normal abelian subgroup of bounded index, or that |G0| < 6.

36 5.2 Further Work

In the following section, we discuss ways in which Hegarty’s work may be extended to prove Joseph’s conjectures for a wider range, or even for the whole of (0, 1].

5.2.1 Groups with |G0| = 8. The most obvious way to improve Hegarty’s work would be to show that the conjectures hold for groups such that |G0| = 8. 0 ∼ 0 ∼ From Section 2.2.3 there are no groups for which G = D8 and if G = C8 0 ∼ then Z(G) must be even. Furthermore, Z(Q8) = {±1} so if G = Q8, Z(G) = {±1}. Thus the cases which must be checked are:

0 3 G C8 C2 × C4 C2 Q8 Z(G) C2 {e}{e} {±1} C4 C2 C2 2 C8 C4 C2 2 3 C2 C2 C2 × C4

Of course, in some cases we may use the same reasoning as in the case |G0| < 8. If Z(G) = {e} then again, by Proposition 4.7, G may be isomorphic to only a finite number of groups. Thus Pr(G) may take only finitely many values, so will satisfy Joseph’s conjectures. If G0 = Z(G), then G is nilpotent so is the product of its Sylow subgroups. In this case, G0 is a 2-group so the only non-abelian Sylow subgroup of G will be its 2-Sylow with commutator subgroup equal to G0. Thus   1 X [G0 : K] Pr(G) = 1 +  8 22n(K)+1 K≤G0 where, as usual, the sum is taken over all strict subgroups K of H0 such that H0/K is cyclic and the integer n(K) is defined as

|H| pn(K) := . |K∗| If |G0| = 8 and |Z(G)| = 4 then [G0 : Z(G)] = 2. Moreover, trivially, gcd(1, |G|) = 1 so G satisﬁes the conditions of Proposition 4.12 and is nilpotent. Then, as in the case G0 = Z(G), we have   1 X [G0 : K] Pr(G) = 1 +  . 8 22n(K)+1 K≤G0

37 However we have been unable to develop any meaningful results for the case |G0| = 8 and |Z(G)| = 4. Moreover, such a result would lead to a proof of Joseph’s conjectures over a range only slighter larger than that in Hegarty’s proof. Thus, ﬁnding a more general result would be a much more eﬀective approach that continuing to check cases for |G0| = 8. Such a result is discussed in the next section.

5.2.2 A General Formula for the Case |G0| ≤ n. In his concluding remarks, Hegarty suggests that a useful next step would be to show that Joseph’s conjectures hold for

0 Cn := {G ∈ G : |G | ≤ n} where n ∈ N is fixed. This would automatically treat the case |G0| ≤ 8 and may also be of use in proving that Joseph’s conjectures hold for all limit points of Pr(G). In this section we develop a formula for the commuting probability of a group with |G0| ≤ n for n ∈ N fixed which we believe may be of use in proving that Pr(Cn) is a good set. We first prove a generalisation of Theorem 4.16:

Theorem 5.4. If H E G and [G : H] = k then Pr(H) 1 X Pr(G) = + (|C (x)| + |C (x)|) k2 |G|2 G H x∈G−H

where CH (x) = CG(x) ∩ H. Proof. First let

K = {(a, b): a, b ∈ G, ab = ba},

K0 = {(a, b) ∈ K : a, b ∈ H},

K1 = {(a, b) ∈ K : a ∈ H, b ∈ G − H},

K2 = {(a, b) ∈ K : a ∈ G − H, b ∈ G}. So that |K| |K | Pr(G) = and Pr(H) = 0 . |G|2 |H|2 Also, X X |K1| = |CH (x)| and |K2| = |CG(x)|. x∈G−H x∈G−H Finally, X |K| = |K0| + |K1| + |K2| = |K0| + (|CH (x)| + |CG(x)|). x∈G−H

38 Then dividing through by |G|2 gives the desired result. Theorem 5.5. If G is a ﬁnite group such that |G0| = n then there exists an integer k ≤ n! such that

k 0 !! n nk 1 Y 1 X (p − 1)[H : K] X |Li| X |Mi| Pr(G) = 1 + i + + . k2 |H0| pn(K)+1 i|G| i|G| i=1 i K i=1 i=1 Where

0 • The ﬁrst sum is over K such that Hi/K is cyclic and the integer n(K) is n(K) ∗ deﬁned as p := [Hi : K ].

• The set Li is deﬁned as Li := {x ∈ G − H : |G|/|CG(x)| = i}.

• The set Mi is deﬁned as Mi := {x ∈ G − H : |G|/|CH (x)| = i}.

0 0 Proof. Suppose that G ∈ Cn. As |G | = n, |Aut(G )| ≤ n!. So, from Lemma 0 4.11, CG(G ) is a normal subgroup of G of ﬁxed index less than or equal to n!. 0 Now let H := CG(G ) and k := [G : H]. By Theorem 5.4,

Pr(H) 1 X Pr(G) = + (|C (x)| + |C (x)|) k2 |G|2 G H x∈G−H

Now if x ∈ G − H,

1 ≤ [G : CG(x)] = |ClG(x)| = |[G, x]| ≤ n.

So |C (x)| G ∈ {1, 1/2, 1/3,..., 1/n}. |G| Moreover, C (x) C (x) HC (x) G G = G =∼ G ≤ . CH (x) H ∩ Cg(x) H H

So [CG(x): CH (x)] divides [G : H] = k ≤ n! and |C (x)| |C (x)| |C (x)| H = H · G ∈ {1, 1/2, 1/3,..., 1/nk}. |G| |CG(x)| |G| For 1 ≤ i ≤ n and 1 ≤ j ≤ nk, let

Li = {x ∈ G − H : |G|/|CG(x)| = i} and Mj = {x ∈ G − H : |G|/|CH (x)| = j}.

39 Then n nk Pr(H) X |Li| X |Mi| Pr(G) = + + . k2 i|G| i|G| i=1 i=1 From Lemma 2.3, H is nilpotent so is the product of its Sylow subgroups. 0 0 0 If H1,...,Hj are its Sylow subgroups then H = H1 × ... × Hj so a p-Sylow of 0 H is non-abelian iﬀ p divides |H |. Thus if p1, . . . , pl are the prime divisors of 0 |H | and H1,...,Hl are the respective pi-Sylows, then from Rusin’s equation,

k !! Y 1 X (p − 1)[H0 : K] Pr(H) = 1 + i . |H0| pn(K)+1 i=1 i K Where there are only a ﬁnite number of possibilities for K (and thus for 0 [Hi : K]).

At this stage, this formula is not suﬃcient to show that Pr(Cn) is a good set. The term corresponding to Pr(H)/k2 can be written in the form

n X 1 x i=1 i

for n ∈ N ﬁxed and xi ∈ Z. However, the i|G|/|Li| and the i|G|/|Mi| are not necessarily integers, so this is not true of the whole expression. However, this formula does have some useful properties. Most importantly, from the deﬁnition of the Li and the fact that H is a subgroup of index k,

|Li| ≤ |G| − |H| = |G|(1 − 1/k).

So the |Li|/|G| are bounded from above, as are the Mi. This alone is not suﬃcient to show that Pr(Cn) is a good set. However, if we write

n nk Pr(H) X ai X ci Pr(G) = + + . k2 b d i=1 i i=1 i

where gcd(ai, bi) = gcd(ci, di) = 1 then if we could show that there were only finitely many values that the ai and the ci could take, then we would be done. It is not clear whether this can be achieved but there is certainly a lot more which can be said about the terms |Li|/|G| and |Mi|/|G|. For example, note that the elements of Li are the elements x ∈ G such that |ClG(x)| = i and the elements of Mi are the elements x ∈ G such that the orbit of x under the action of H is of size i. Thus i divides both Li and Mi. We believe that, given further study of the Li and the Mi, this formula may be used to show that Joseph’s conjectures are satisfied by groups G for which |G0| obeys a fixed upper bound.

40 Chapter 6

Concluding Remarks

It is clear that there is still much which can be said about the limit points of the commuting probability function on finite groups. Although some progress may be made by adjusting certain stages in Hegarty’s proof, there are limited improvements which can be made using this approach. Thus, in order to prove Joseph’s first and second conjectures for all limit points of Pr(G), it is likely that a completely new method is required. 0 Hegarty [12] suggests that showing that Pr(Cn) (where Cn = {G ∈ G : |G | ≤ n}) is a good set may form one of the steps in this. We hope that our formula will go some way to proving this, or will at least give further insight into the limit points of Pr(Cn). As for the third conjecture, our conclusion is the much same as that of Hegarty. This statement seems much more mysterious than the first two conjectures and our investigation into the subject has given us no more insight into how this may be proved. Finally, it is also worth noting that the methods used by both Rusin [25] and Das and Nath [5] rely on finding explicit formulae for the commuting probabilities of groups with small commutator subgroups. Thus, with some refinement, our formula may also help towards extending the results of both these papers.

Forwarding Address: Madeleine Whybrow St Catherine Cliﬀ Road Hythe Kent, CT21 5XW

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