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 (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 jG0j < 8..................... 33 5.1.2 The Main Proof . 35 5.2 Further Work . 37 5.2.1 Groups with jG0j = 8. 37 5.2.2 A General Formula for the Case jG0j ≤ n. 38 6 Concluding Remarks 41 1 Chapter 1 Introduction Let G be the set of all finite groups then, for G 2 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 jf(x; y) 2 G × G : xy = yxgj Pr(G) = : jGj2 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 bound- ing 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) [ f0g is a closed subset of R. 1 For our purpose, we say that l 2 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 2 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 2 N be fixed then ( n ) X 1 S := : x 2 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 sub- group, 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 jG0j where d is the smallest degree of a non-linear complex irreducible character of G. In fact, when d = 3 and jG0j = 8, 1 1 1 2 + 1 − = : d2 d2 jG0j 9 2 0 So if Pr(G) > 9 then either jG j < 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; jG0j = 9 + 1 − = 81 d2 d2 jG0j 23 0 128 ≈ 0:1797 if d = 4; jG j = 8: So, in order to prove the conjectures over a wider range, we must first prove that the conjectures are true when jG0j < 9. The case jG0j < 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 2 X; y 2 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 2 G such that [x; y] 2= 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 8x; y 2 G,[x; y] 2 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 sub- group: 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.