CENTRAL CAMINA PAIRS a Dissertation Submitted to Kent State

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CENTRAL CAMINA PAIRS

A dissertation submitted to Kent State University in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

by David G. Costanzo

May 2020

c Copyright All rights reserved Except for previously published materials Dissertation written by David G. Costanzo B.S., University of Scranton, 2010 M.S., University of Scranton, 2012 M.S., Kent State University, 2018 Ph.D., Kent State University, 2020

Approved by

Mark L. Lewis , Chair, Doctoral Dissertation Committee

Stephen M. Gagola, Jr. , Members, Doctoral Dissertation Committee

Donald L. White

Scott A. Courtney

Joanne Caniglia

Accepted by

Andrew M. Tonge , Chair, Department of Mathematical Sciences

Mandy Munro-Stasiuk, Ph.D. , Interim Dean, College of Arts and Sciences TABLE OF CONTENTS

TABLE OF CONTENTS ...... iii

ACKNOWLEDGMENTS ...... v

1 INTRODUCTION ...... 1

2 ELEMENTARY GROUP THEORY ...... 6

3 CAMINA PAIRS ...... 12 3.1 Camina Pairs ...... 12 3.2 Central Camina Pairs ...... 14

4 MAIN RESULTS ON CENTRAL CAMINA PAIRS ...... 17 4.1 Central Camina Pairs of Nilpotence Class at Least 4 ...... 17 4.2 Central Camina Pairs of Nilpotence Class 3 ...... 20 4.3 Resolving Conjecture (L) in Special Cases ...... 21 4.4 A Minimal Counterexample to Conjecture (L) ...... 22 4.5 More Results on Central Camina Pairs ...... 26

5 A PARTICULAR GRAPH ATTACHED TO A GROUP . . . . . 30 5.1 Introduction ...... 30 5.2 More Elementary Group Theory ...... 31 5.3 Basic Graph Theory ...... 32 5.4 Preliminary Observations ...... 33 5.5 2-Frobenius Groups ...... 36

BIBLIOGRAPHY ...... 42

iii DEDICATION

This work is dedicated to the memory of my father Gabriel R. Costanzo.

iv ACKNOWLEDGMENTS

I thank my advisor, Dr. Mark L. Lewis. I thank him for formulating the conjecture that so greatly stimulated my interest. I thank him for his patience as I was presenting trial runs in an attempt to resolve this conjecture. Finally, I thank him for playing such a important role in my mathematical development. Many thanks go to Dr. Donald L. White, whose character theory course in the Spring 2018 semester beneﬁted me greatly. I also thank Dr. White for carefully reading this dissertation and providing me with excellent feedback, which led to many improvements. I thank Dr. Stephen M. Gagola, Jr. for many useful comments on a draft of this dissertation and, most importantly, I thank him for Theorem 6.3 in [3]. I thank Stefano Schmidt, Eyob Tsegaye, and Gabe Udell for an awesome summer of research. I am extremely grateful that our paths crossed. I thank Alexander D. Bongiovanni for many conversations about the graduate student experience. Many thanks go to Edward D. Penetar for years and years of unwavering friendship. I also thank him for his countless attempts to keep me sane. Finally, I thank my mother, Susan A. Costanzo, for her support and patience throughout this process.

v CHAPTER 1

INTRODUCTION

All groups under consideration are assumed to be of ﬁnite order. The primary focus of this dissertation is central Camina pairs. In the last chapter, however, the cyclic graph of a 2-Frobenius group is discussed. Let G be a group. Assume that G has a nontrivial proper subgroup H satisfying H ∩ Hg = { 1 } for every g ∈ G \ H. Under this assumption, G is called a Frobenius group and H is called a Frobenius complement. If G is a Frobenius group with Frobenius complement H, then the set ! [ N = G \ Hg ∪ { 1 } g∈G

turns out to be a normal subgroup of G. The subgroup N is called the Frobenius kernel of G. Let G be a Frobenius group with Frobenius kernel N, and consider the following condition, which is satisﬁed by the elements in N and the elements in G \ N.

For every x ∈ G \ N and for every a ∈ N, the element x is conjugate to xa.

A.R. Camina [1] studied the structure of a group G with a nontrivial proper normal subgroup N satisfying this condition. One of the main results obtained in [1] is the following classiﬁcation.

Theorem (Camina, [1], Theorem 2). Let G be a group and let N be a nontrivial proper normal subgroup of G. Assume that for every x ∈ G \ N and for every a ∈ N, the element x is conjugate to xa. Then

1 1. G is a Frobenius group with Frobenius kernel N, or

2. G/N or N is a p-group for some prime p.

Several authors continued this line of work, and the following definition became standard. If G is a group with a nontrivial proper normal subgroup N such that for every x ∈ G \ N and for every a ∈ N, the element x is conjugate to xa, then (G, N) is called a Camina pair. The notion of a Camina pair has numerous equivalent formulations; see Lemma 3.1 in this dissertation and Lemma 4.1 in [10]. In particular, if N is a nontrivial proper normal subgroup of G, then (G, N) is a Camina pair if and only if for every x ∈ G \ N and for every a ∈ N, there exists an element b ∈ G such that a = [b, x]. This condition shall be our official definition of a Camina pair. Let (G, N) be a Camina pair. Then Z(G) ≤ N ≤ G0; see Lemma 3.2. If N = G0, then G is called a Camina group. Camina groups have been the subject of much investigation; see [2], [9], [11], and [12]. The focus in this dissertation is the other extreme: namely, N = Z(G). A Camina pair (G, Z(G)) is called a central Camina pair. Mark L. Lewis pioneered an investigation of central Camina pairs in [8]. Lewis made the following key observation.

Theorem (Lewis, [8], Lemma 2.1). If (G, Z(G)) is a Camina pair, then G is a group of prime power order.

The proof given in [8] of the previous fact uses Camina’s classiﬁcation result mentioned previously. A proof that does not rely on this classiﬁcation appears below as Lemma 3.5. The following conjecture was made in [8].

Conjecture (L). If (G, Z(G)) is a Camina pair, then |G : Z(G)| ≥ |Z(G)|2.

2 Lewis was able to obtain the following bound, which appears as Theorem 3 in [8].

Theorem (Lewis, [8], Theorem 3). If (G, Z(G)) is a Camina pair, then

|G : Z(G)| > |Z(G)|4/3.

Let (G, Z(G)) be a Camina pair, and recall that G must be a p-group for some prime p. Under the additional condition that the factor group G/Z(G) has exponent at least p2, Lewis was indeed able to establish that |G : Z(G)| ≥ |Z(G)|2; in fact, a stronger bound is obtained. The following theorem is Theorem 5 in [8].

Theorem (Lewis, [8], Theorem 5). If (G, Z(G)) is a Camina pair where G is a p-group and the factor group G/Z(G) has exponent pn, n ≥ 2, then

|G : Z(G)| ≥ |Z(G)|npn.

Resolving Conjecture (L) is the primary motivation of this dissertation. In contrast to the eﬀorts mentioned above, the approach here is to partition the problem according to the nilpotence class of G. In particular, Conjecture (L) is resolved when the nilpotence class of G is at least 4.

Theorem A. If (G, Z(G)) is a Camina pair where the group G has nilpotence class at least 4, then |G : Z(G)| > |Z(G)|2.

Caution should be exercised here. If specialized conditions are imposed on a nilpotent group, then there may exist a constant that bounds the nilpotence class. In particular, the condition that (G, Z(G)) forms a Camina pair may force G to have a small nilpotence class, which of course lessen the content of Theorem A. Fortunately, central Camina pairs of nilpotence class at least 4 are abundant. The following theorem recasts Theorem 6.3 in [3] in our present terminology.

3 Theorem (Gagola, [3], Theorem 6.3). If P is a p-group, then there exists a group G such that (G, Z(G)) is a Camina pair and P is isomorphic to a subgroup of G/Z(G).

In particular, notice that Gagola’s result yields that central Camina pairs can have arbitrarily large nilpotence class. Let (G, Z(G)) be a Camina pair. With Conjecture (L) resolved when G has nilpotence class at least 4, the situation where G has nilpotence class 3 needs to be handled. Unfortunately, Conjecture (L) remains open under this hypothesis. The following inequality, however, is obtained; note that this inequality improves existing bounds.

Theorem B. If (G, Z(G)) is a Camina pair where the group G has nilpotence class 3, then |G : Z(G)| > |Z(G)|3/2.

Let (G, Z(G)) be a Camina pair and assume that G has nilpotence class 2. Under these hypotheses, G0 = Z(G), and so G is a Camina p-group of nilpotence class 2. I.D. MacDonald [11] proved that |G : Z(G)| ≥ |Z(G)|2 in this situation. Combining this fact with the results above, the following general bound for central Camina pairs is obtained.

Corollary C. If (G, Z(G)) is a Camina pair, then

|G : Z(G)| > |Z(G)|3/2.

0 Seeking another way to approach Conjecture (L), recall that Z2(G) ≤ CG(G ) for

0 0 any group G. So, if G is a group, then either Z2(G) = CG(G ) or Z2(G) < CG(G ). A strong bound is obtained when (G, Z(G)) is a Camina pair satisfying the condition

0 that Z2(G) < CG(G ).

0 Theorem D. If (G, Z(G)) is a Camina pair with Z2(G) < CG(G ), then

|G : Z(G)| ≥ |Z(G)|2|G0 : Z(G)|.

4 As a consequence of Theorem D, the inequality |G : Z(G)| ≥ |Z(G)|2 holds when

0 (G, Z(G)) is a Camina pair with Z2(G) < CG(G ). Hence Conjecture (L) is also resolved under this additional hypothesis. Given the hypotheses of Theorem D, the group G has nilpotence class at least 3. Thus Z(G) < G0, and so the strict inequality |G : Z(G)| > |Z(G)|2 holds. In central Camina pairs, the hypotheses that the group G has nilpotence class at

0 least 4 and that Z2(G) < CG(G ) yield the following bound.

Corollary E. If (G, Z(G)) is a Camina pair, the group G has nilpotence class at least

0 4, and Z2(G) < CG(G ), then

|G : Z(G)| > |Z(G)|3.

Let (G, Z(G)) be a Camina pair. In light of the previous results, Conjecture (L) is now only open when G has nilpotence class 3, the factor group G/Z(G) has exponent

0 p, and Z2(G) = CG(G ). In Section 4.4 of this dissertation, however, some work is presented toward remedying this situation.

5 CHAPTER 2

ELEMENTARY GROUP THEORY

A group is a set G together with an associative binary operation ◦ such that

• (Existence of an Identity) there exists an element 1 ∈ G with the property that 1 ◦ g = g ◦ 1 = g for all g ∈ G, and

• (Existence of Inverses) for each element g ∈ G, there exists an element g−1 ∈ G such that g ◦ g−1 = g−1 ◦ g = 1.

It is customary to suppress the symbol for the binary operation and, instead, to juxtapose group elements. The cardinality of the underlying set of a group is called the order of the group. If G is a group, then |G| denotes its order.

In this dissertation, all groups are assumed to be of ﬁnite order.

Let G be a group. A subset H of G is said to be a subgroup of G if H is itself a group upon restricting the binary operation of G to H. The notation H ≤ G is used to indicate that H is a subgroup of G. Let H ≤ G, and let x ∈ G. The set Hx = { hx | h ∈ H } is called a right coset of H in G. The set xH = { xh | h ∈ H } is called a left coset of H in G. The term coset should be interpreted as right coset. Let G be a group, and let H ≤ G. Let Ω = { Hx | x ∈ H }, the set of right cosets of H in G, and let Ω0 = { xH | x ∈ H }, the set of left cosets of H in G. Consider the correspondence f from Ω into Ω0 given by f : Hx 7→ x−1H. Then, f is well-deﬁned

6 and constitutes a bijection from Ω onto Ω0. Hence, the number of right cosets of H in G is equal to the number of left cosets of H in G. Write |Ω| = |Ω0| = |G : H|. The positive integer |G : H| is called the index of H in G. Let G be a group, and let N ≤ G. If Ng = gN for each g ∈ G, then N is said

to be a normal subgroup of G. The notation N E G indicates that N is a normal subgroup of G. Let N E G. A binary operation on the set of all cosets of N in G is deﬁned by NxNy = Nxy (x, y ∈ G).

The coset space { Nx | x ∈ G } equipped with this binary operation forms a group. This group is denoted G/N and is called the factor group of G with respect to N. Let G and H be groups. A homomorphism is a map σ : G → H such that

(xy)σ = (x)σ(y)σ

for every x, y ∈ G. If, in addition to being a homomorphism, σ is a bijection, then it is called an isomorphism, and the groups G and H are said to be isomorphic.

Let G be a group, and let X be a nonempty subset of G. The set

CG(X) = { g ∈ G | gx = xg for each x ∈ X }

is called the centralizer of X in G. For g ∈ G, let Xg = { g−1xg | x ∈ X }. The set Xg is called a conjugate of X. The set

g NG(X) = { g ∈ G | X = X}

is called the normalizer of X in G. Note that CG(X) and NG(X) form subgroups of

G. If Y ⊆ CG(X), then Y is said to centralize X and, similarly, if Y ⊆ NG(X), then

Y is said to normalize X. If X = {x}, then write CG(X) = CG(x). Let G be a group. Put

Z(G) = { z ∈ G | zg = gz for each g ∈ G }.

7 The set Z(G) forms a subgroup of G called the center of G. Notice that Z(G) is a normal subgroup of G.

The following two lemmas concern subgroups and indices. They are quite elementary, but they will be used proﬁtably in upcoming chapters.

Lemma 2.1 (Dedekind’s Lemma). If H ≤ K ≤ G and L ≤ G, then

K ∩ HL = H(K ∩ L).

Lemma 2.2. If G is a group and H,K ≤ G, then |G : H ∩ K| ≤ |G : H||G : K|.

Let G be a group, and let Ω be a nonempty set. Assume that for each g ∈ G and for each α ∈ Ω, an element αg ∈ Ω is deﬁned. Furthermore, suppose that

1. α1 = α for each α ∈ Ω, and

2. (αg)h = αgh for each α ∈ Ω and for each g, h ∈ G.

Then G is said to act on Ω. When a group G acts on a set Ω, the elements in Ω are usually called points. Let G be a group, and suppose that G acts on the set Ω. Let α ∈ Ω. The set

αG = { αg | g ∈ G }

is the G-orbit of α. It is easy to check that the G-orbits partition Ω. The set

g Gα = { g ∈ G | α = α }

is the point-stabilizer of α. It is routine to verify that Gα forms a subgroup of G. The following result is one of the most fundamental results regarding group actions.

Theorem 2.3 (Orbit-Stabilizer Theorem). Let G be a group acting a set Ω, and let ∆ be a G-orbit. Let α be any point in ∆. Then

|∆| = |G : Gα|.

8 Let G be a group, and let Ω be the set of all nonempty subsets of G. Then G acts on Ω by conjugation. If X ∈ Ω, then the G-orbit of X is called the conjugacy class of

X. The stabilizer of X with respect to this action is NG(X), and so

G |X | = |G : NG(X)|.

G If X = {x}, then |x | = |G : CG(x)|.

Let G be a group, and let x, y ∈ G. The element [x, y] = x−1y−1xy is called the commutator of x and y. For x, y, z ∈ G, write [[x, y], z] = [x, y, z]. The following lemma lists commutator identities. The veriﬁcation of these identities is straight-forward.

Lemma 2.4 (Commutator Identities). Let G be a group, and let x, y, z ∈ G. Then,

1. [x, y]−1 = [y, x]

2. [xy, z] = [x, z]y[y, z]

3. [x, yz] = [x, z][x, y]z

4. [xy, z] = [x, z][x, z, y][y, z]

5. [x, yz] = [x, z][x, y][x, y, z].

The following commutator identity plays a crucial role in this dissertation.

Lemma 2.5 (The Hall-Witt Identity). If G is a group and x, y, z ∈ G, then

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

Let G be a group, and let X and Y be nonempty subsets of G. The subgroup

[X,Y ] = h [x, y] | x ∈ X, y ∈ Y i

9 is called the commutator of X and Y . In the special case where X = Y = G, the familiar derived subgroup G0 is obtained. For nonempty subsets X, Y , and Z of G, write [[X,Y ],Z] = [X,Y,Z]. The following lemmas gather together the basic results concerning commutator subgroups that are needed in later chapters.

Lemma 2.6. If G is a group and H,K ≤ G, then [H,K] = [K,H].

The next result, which is a consequence of the Hall-Witt Identity, is extremely useful.

Lemma 2.7 (Three Subgroups Lemma). Let G be a group, N be a normal subgroup of G, and X,Y,Z ≤ G. If [X,Y,Z], [Y,Z,X] ≤ N, then [Z,Y,X] ≤ N.

The special case N = { 1 } yields the following result.

Lemma 2.8. Let G be a group, and let X, Y , and Z be subgroups of G. If [X,Y,Z] = [Y,Z,X] = { 1 }, then [Z,X,Y ] = { 1 }.

Let G be a group. Let Z0(G) = { 1 }, and for i ≥ 1, let Zi(G)/Zi−1(G) =

Z(G/Zi−1(G)). The series

{ 1 } = Z0(G) ≤ Z1(G) = Z(G) ≤ Z2(G) ≤ · · ·

is called the upper central series of G. The term Zi(G) is called the ith center of G.

Next, let G1 = G, and for i ≥ 2, set Gi = [Gi−1,G]. Then,

0 G = G1 ≥ G2 = G ≥ · · ·

is called the lower central series of G. Let G be a group. If the upper central series reaches all of G, then G is said to be nilpotent. For a nilpotent group G, the least nonnegative integer c such that

Zc(G) = G is called the nilpotence class of G, or sometimes just the class of G.

10 The following fact is a fundamental result concerning nilpotent groups; it says that nontrivial normal subgroups of a nilpotent group intersect the center nontrivially.

Lemma 2.9. If G is a nilpotent group and N is a nontrivial normal subgroup of G, then N ∩ Z(G) 6= { 1 }.

Let p be a prime number. A group G of order pn, where n is a nonnegative integer, is a called a p-group. Note that groups of prime power order are nilpotent. The proofs of the following two lemmas are easy applications of the Three Subgroups Lemma.

0 Lemma 2.10. If G is a group, then Z2(G) ≤ CG(G ).

0 Lemma 2.11. If G is a group, then [G , Z3(G)] ≤ Z(G).

The following lemma delivers a count on commutators of a particular form.

Lemma 2.12. If G is a group, H is a subgroup of G, and x ∈ G, then

|{ [h, x] | h ∈ H }| = |H : CH (x)|.

A group G is called a Frobenius group if it has a nontrivial proper subgroup H satisfying the condition that H ∩ Hg = { 1 } for each g ∈ G \ H. The subgroup H just described is called a Frobenius complement. If G is a Frobenius group with Frobenius complement H, then the set ! [ N = G \ Hg ∪ { 1 } g∈G

turns out to be a normal subgroup of G. The subgroup N is called the Frobenius kernel of G.

11 CHAPTER 3

CAMINA PAIRS

3.1 Camina Pairs

Let G be a group and let N be a nontrivial proper normal subgroup of G. The pair (G, N) is called a Camina pair if for every x ∈ G \ N and for every a ∈ N, there exists an element b ∈ G such that a = [b, x]. The subgroup N is called a Camina kernel. Evidently, if G is a Frobenius group with Frobenius kernel N, then (G, N) is a Camina pair. Let p be a prime. Recall that a p-group G is said to be extraspecial if Z(G) = G0 = Φ(G) and |Z(G)| = p. (Here, as usual, Φ(G) is denoting the Frattini subgroup of G.) If G is an extraspecial p-group, then (G, Z(G)) is a Camina pair. Hence the extraspecial p-groups furnish another family of examples of Camina pairs. The following lemma lists a few equivalent conditions.

Lemma 3.1. Let G be a group, and let N be a nontrivial proper normal subgroup of G. The following conditions are equivalent:

1. (G, N) is a Camina pair,

2. for every x ∈ G \ N and for every a ∈ N, the element x is conjugate to xa,

3. for every x ∈ G \ N, the equality |CG(x)| = |CG/N (Nx)| holds.

A.R. Camina’s initial investigation of groups satisfying this condition yielded the following classiﬁcation result.

Theorem (Camina, [1], Theorem 2). If (G, N) is a Camina pair, then

1. G is a Frobenius group with Frobenius kernel N, or

12 2. G/N or N is a p-group for some prime p.

Lemma 3.2 provides information about the location of a Camina kernel.

Lemma 3.2. If (G, N) is a Camina pair, then Z(G) ≤ N ≤ G0.

Proof. The inclusion N ≤ G0 follows from deﬁnition. Aiming for a contradiction, assume that there exists z ∈ Z(G)\N. Fix a ∈ N #. By the Camina condition, a = [b, z] for some b ∈ G. But z is central, and so a = [b, z] = 1,

a contradiction. Hence Z(G) ≤ N.

If G is nilpotent, then more precise information about the location of a Camina kernel is known.

Theorem 3.3 (MacDonald, [11], Lemma 2.1). Let (G, N) be a Camina pair. If G is nilpotent, then N is simultaneously a term of the upper and lower central series of G.

A new Camina pair can be built from an existing one, as described in the following lemma.

Lemma 3.4. If (G, N) is a Camina pair and K is a normal subgroup of G that is properly contained in N, then (G/K, N/K) is a Camina pair.

Proof. The hypothesis and the Correspondence Theorem give us that N/K is a nontrivial proper normal subgroup of G/K. Thus, the Camina condition must be established. Let Kx ∈ (G/K) \ (N/K) and Ka ∈ N/K. Then, x ∈ G \ N and a ∈ N. Because (G, N) forms a Camina pair, there exists some element b ∈ G such that a = [b, x]. Thus, Ka = K[b, x] = [Kb, Kx].

The elements Kx ∈ (G/K) \ (N/K) and Ka ∈ N/K were arbitrary, and so the pair

(G/K, N/K) is a Camina pair.

13 3.2 Central Camina Pairs

Recall that a central Camina pair is a Camina pair of the form (G, Z(G)). In other words, a central Camina pair is a Camina pair where Z(G) is a Camina kernel. It turns out that if (G, Z(G)) is a Camina pair, then G is a group of prime power order; this fact appears as Lemma 2.1 in [8]. The proof given there uses a classiﬁcation result due to A.R. Camina. The proof below is self-contained.

Lemma 3.5. If (G, Z(G)) is a Camina pair, then G is a p-group for some prime p.

Proof. Camina kernels are proper subgroups, and so Z(G) cannot contain every Sylow

subgroup of G. Thus, there is some prime p and some P ∈ Sylp(G) such that P 6≤ Z(G). Fix x ∈ P \ Z(G), and let z ∈ Z(G) be arbitrary. By the Camina condition, there exists some y ∈ G such that z = [y, x]. Now [y, x] = z ∈ Z(G), and so, using basic commutator identities, zo(x) = [y, x]o(x) = [y, xo(x)] = [y, 1] = 1. Hence o(z) divides o(x). Since x ∈ P , the integer o(x) is a power of p. It follows that o(z) is also a power of p. Because z ∈ Z(G) was arbitrary, every element in Z(G) has order a power of p. Thus, Z(G) is a p-group. Now Z(G) is a normal p-subgroup of G, and so Z(G) ≤ P . Suppose that G is not a p-group, and let q 6= p be a prime divisor of |G|. Let u ∈ G with o(u) = q. Since Z(G) is a p-group, u∈ / Z(G). Choose z ∈ Z(G)# = Z(G) \{ 1 }. By the Camina condition, there exists some y ∈ G such that z = [y, u]. Since [y, u] ∈ Z(G), it follows that zq = [y, u]q = [y, uq] = 1. Thus o(z) divides q. But o(z)

is a power of p. This situation forces z = 1, contrary to the choice of z.

If (G, Z(G)) forms a Camina pair, then the structure of Z(G) is determined; in par-

ticular, Z(G) is an elementary abelian p-group. Moreover, each factor Zi(G)/Zi−1(G) of the upper central series is elementary abelian.

Theorem 3.6 (MacDonald, [11], Theorem 2.2). Let (G, Z(G)) be a Camina pair, and write c for the nilpotence class of G. Then Zi(G)/Zi−1(G) has exponent p for each i = 1, . . . , c.

14 If (G, Z(G)) is a Camina pair, then information about |G : Z(G)| is available. The next lemma is Lemma 2.4 in [8].

Lemma 3.7 (Lewis, [8], Lemma 2.4). If (G, Z(G)) is a Camina pair, then |G : Z(G)| is a square.

Let G be a group, and let x ∈ G. Deﬁne

DG(x) = { g ∈ G | [g, x] ∈ Z(G) }.

Note that DG(x) forms a subgroup of G; more speciﬁcally,

DG(x)/Z(G) = CG/Z(G)(Z(G)x).

The following lemma is crucial to the forthcoming work.

Lemma 3.8 (Lewis, [8], Lemma 3.2). If (G, Z(G)) is a Camina pair and x ∈ G\Z(G), then CG(x) E DG(x) and ∼ DG(x)/CG(x) = Z(G).

In particular, |DG(x): CG(x)| = |Z(G)| for every x ∈ G \ Z(G).

If (G, N) is a Camina pair and K is a normal subgroup of G with K < N, then (G/K, N/K) is a Camina pair, see Lemma 3.4. This fact describes a way to build a new Camina pair from an existing one. The following result describes a way to build a new central Camina pair from an existing one.

Lemma 3.9. If (G, Z(G)) is a Camina pair and Z < Z(G), then (G/Z, Z(G/Z)) is a Camina pair.

Proof. Given these assumptions, (G/Z, Z(G)/Z) is a Camina pair. Since Z(G)/Z is a Camina kernel, Z(G)/Z ≥ Z(G/Z). The inclusion Z(G)/Z ≤ Z(G/Z) also holds.

Hence Z(G)/Z = Z(G/Z).

15 Let G be a group, and let H ≤ G. For x ∈ G, set [H, x] = h [h, x] | h ∈ H i. (This notation is consistent with the notation introduced in Chapter 2 since [H, x] = [H, { x }]. Note that sometimes authors write [H, x] for the set of commutators of the form [h, x] with h ∈ H.)

Lemma 3.10. If (G, Z(G)) is a Camina pair, then Z(G) = [DG(x), x] for every x ∈ G \ Z(G).

Proof. Let x ∈ G \ Z(G). Each commutator of the form [d, x], with d ∈ DG(x),

belongs to Z(G). As the set { [d, x] | d ∈ DG(x) } generates [DG(x), x], it follows that

[DG(x), x] ≤ Z(G). Next, take z ∈ Z(G) and, using the Camina condition, write

z = [d, x] for some d ∈ G. Then d ∈ DG(x) by deﬁnition, and so z ∈ [DG(x), x]. Now

Z(G) ≤ [DG(x), x] since z was arbitrary.

Lemma 3.11. If (G, Z(G)) is a Camina pair, then |G : G0| ≥ |Z(G)|.

0 Proof. Recall that [Z2(G),G ] = { 1 }. Let x ∈ Z2(G) \ Z(G). Note that DG(x) = G,

0 0 and so |G : CG(x)| = |Z(G)|. Thus |G : G | ≥ |Z(G)| since G ≤ CG(x).

16 CHAPTER 4

MAIN RESULTS ON CENTRAL CAMINA PAIRS

4.1 Central Camina Pairs of Nilpotence Class at Least 4

Let (G, Z(G)) be a Camina pair where the group G has nilpotence class at least 4. In Theorem 4.3, the inequality |G : Z(G)| > |Z(G)|2 is established, and so Conjecture (L) is completely resolved under this additional assumption on the nilpotence class of G. Before proceeding any further, however, a result that ensures Theorem 4.3 has content must be mentioned.

Theorem (Gagola, [3], Theorem 6.3). If P is a p-group, then there exists a group G such that (G, Z(G)) is a Camina pair and P is isomorphic to a subgroup of G/Z(G).

Consequently, there exist central Camina pairs with arbitrarily large nilpotence class. The proof of Theorem 4.3 relies on the fact that a particular set of elements in a group with nilpotence class at least four is nonempty.

Lemma 4.1. If G is a nilpotent group with nilpotence class at least 4, then the set

0 (G ∩ Z3(G)) \ Z2(G) is nonempty.

0 0 Proof. Since G 6≤ Z2(G), the subgroup G Z2(G)/Z2(G) is a nontrivial normal sub-

0 group of the factor group G/Z2(G). Thus G Z2(G)/Z2(G) intersects Z(G/Z2(G)) =

0 Z3(G)/Z2(G) nontrivially. Hence G Z2(G) ∩ Z3(G) > Z2(G). By Dedekind’s Lemma,

0 0 G Z2(G) ∩ Z3(G) = Z2(G)(G ∩ Z3(G)).

17 0 0 Now, if G ∩ Z3(G) ≤ Z2(G), then G Z2(G) ∩ Z3(G) = Z2(G), a contradiction.

0 Therefore, the set (G ∩ Z3(G)) \ Z2(G) is nonempty.

Before proving Theorem 4.3, one more elementary result, which is a consequence of Lemmas 2.11 and 2.12, is posted.

0 Lemma 4.2. If G is a group and x ∈ Z3(G), then [G , x] is a central subgroup of

0 order |G : CG0 (x)|.

Theorem 4.3. If (G, Z(G)) is a Camina pair where the group G has nilpotence class at least 4, then |G : Z(G)| > |Z(G)|2.

0 0 Proof. By Lemma 4.1, (G ∩Z3(G))\Z2(G) is nonempty. Fix x ∈ (G ∩Z3(G))\Z2(G).

0 0 Since x ∈ Z3(G), the commutator [G , x] is a central subgroup of order |G : CG0 (x)|. Set Z = [G0, x].

Suppose that Z = Z(G). Note that Z(G) < CG0 (x) as x ∈ CG0 (x) \ Z(G), and so

0 0 |G : Z(G)| = |G : G ||G : CG0 (x)||CG0 (x): Z(G)|

≥ |Z(G)||Z(G)||CG0 (x): Z(G)|

> |Z(G)|2.

Proceed with the hypothesis that Z < Z(G). By Lemma 3.9, the pair (G/Z, Z(G)/Z) is a Camina pair with Z(G)/Z = Z(G/Z).

Observe that DG(x) < G since x∈ / Z2(G). Let y ∈ G \ DG(x), and note that

−1 −1 [y, x ] ∈/ Z(G) as DG(x) = DG(x ). In the factor group G/Z,

[Zy, Zx−1] = Z[y, x−1] ∈/ Z(G)/Z = Z(G/Z).

−1 Let C be the subgroup of G such that C/Z = CG/Z ([Zy, Zx ]). As (G/Z, Z(G/Z)) is a Camina pair and the element [Zy, Zx−1] is noncentral,

|G : C| = |G/Z : C/Z| ≥ |Z(G/Z)| = |Z(G): Z|.

18 The next task is to show that DG(x) ≤ C. Let d ∈ DG(x). Apply the Hall-Witt Identity to the elements d, x, and y and obtain the equation

[y, x−1, d]x[x, d−1, y]d[d, y−1, x]y = 1.

Note that [x, d−1, y]d = 1 since [x, d−1] ∈ Z(G). It follows that

[y, x−1, d] = (([d, y−1, x]y)−1)x−1 ∈ Z.

Thus d ∈ C. Because d ∈ DG(x) was arbitrary, DG(x) ≤ C. Consequently,

|G : DG(x)| ≥ |G : C| ≥ |Z(G): Z|.

0 The inclusion [G, x] ≤ Z2(G) holds since x ∈ Z3(G). Hence [G, x] ≤ (G ∩Z2(G)) ≤

0 0 0 0 Z(G ). Note that Z(G ) ≤ CG0 (x) as x ∈ G and that (G ∩ Z2(G)) < CG0 (x) since

x∈ / Z2(G). Thus, [G, x] < CG0 (x).

By Lemma 3.10, Z(G) = [DG(x), x], and so Z(G) = [DG(x), x] ≤ [G, x]. The subgroup [G, x] is generated by the set of commutators of the form [g, x], where g ∈ G,

and there are |G : CG(x)| such commutators. In particular,

|[G, x]| ≥ |G : CG(x)| = |G : DG(x)||DG(x): CG(x)| = |G : DG(x)||Z(G)|.

Thus,

|[G, x]: Z(G)| ≥ |G : DG(x)|.

Since [G, x] < CG0 (x),

|CG0 (x): Z(G)| > |[G, x]: Z(G)| ≥ |G : DG(x)| ≥ |Z(G): Z|.

Finally, observe that

0 0 |G : Z(G)| = |G : CG0 (x)||CG0 (x): Z(G)|

= |Z||CG0 (x): Z(G)|

> |Z||Z(G): Z| = |Z(G)|.

0 2 Since |G : G | ≥ |Z(G)|, the inequality |G : Z(G)| > |Z(G)| holds.

19 4.2 Central Camina Pairs of Nilpotence Class 3

Let (G, Z(G)) be a Camina pair where the group G has nilpotence class 3. In this section, the inequality |G : Z(G)| > |Z(G)|3/2 is established. The argument used works as long as G has nilpotence class at least 3. But, as established in the previous section, a much stronger bound is available when G has nilpotence class at least 4.

Lemma 4.4. Let (G, Z(G)) be a Camina pair, and assume that G has nilpotence class at least 3. If x, y ∈ G with [x, y] ∈/ Z(G), then

|G : DG(x) ∩ DG(y)| ≥ |Z(G)|.

Proof. Put U = DG(x) ∩ DG(y), and let u ∈ U. Apply the Hall-Witt Identity to the elements x, y, and u:

[x, y−1, u]y[y, u−1, x]u[u, x−1, y]x = 1.

The second and third factors on the left-hand side are 1 since the commutators [y, u−1] and [u, x−1] are central. Hence [x, y−1, u] = 1. As u ∈ U was arbitrary,

−1 −1 −1 y−1 −1 U ≤ CG([x, y ]). Note that [x, y ] = ([x, y] ) ∈/ Z(G). Now |G : CG([x, y ])| ≥ |Z(G)| since (G, Z(G)) is a Camina pair, and thus the inequality |G : U| ≥ |Z(G)|

holds.

The elements x and y as in the hypothesis of Lemma 4.4 exist in a central Camina

pair (G, Z(G)) where G has nilpotence class at least 3. Take x ∈ G \ Z2(G), and then

take y ∈ G \ DG(x).

Theorem 4.5. If (G, Z(G)) is a Camina pair where the group G has nilpotence class 3, then |G : Z(G)| > |Z(G)|3/2.

Proof. Let x, y ∈ G with [x, y] ∈/ Z(G). Write |Z(G)| = pm. Using Lemma 4.4,

m p ≤ |G : DG(x) ∩ DG(y)| ≤ |G : DG(x)||G : DG(y)|.

Since Z(G) < CG(x),

|G : Z(G)| = |G : DG(x)||DG(x): CG(x)||CG(x): Z(G)|

m/2 m ≥ p p |CG(x): Z(G)|

> pm/2pm = p3m/2 = |Z(G)|3/2.

3/2 So the inequality |G : Z(G)| > |Z(G)| holds.

Corollary C from the Introduction is now immediate.

Corollary 4.6. If (G, Z(G)) is a Camina pair, then

|G : Z(G)| > |Z(G)|3/2.

4.3 Resolving Conjecture (L) in Special Cases

0 0 For any group G, recall that [Z2(G),G ] = { 1 }, and so Z2(G) ≤ CG(G ). Let (G, Z(G)) be a Camina pair. In this section, Conjecture (L) is resolved under the

0 assumption that Z2(G) < CG(G ).

0 Theorem 4.7. If (G, Z(G)) is a Camina pair with Z2(G) < CG(G ), then

|G : Z(G)| ≥ |Z(G)|2|G0 : Z(G)|.

0 −1 Proof. Fix x ∈ CG(G ) \ Z2(G) and y ∈ G \ DG(x). Note that [y, x ] ∈/ Z(G) since

−1 DG(x) = DG(x ). Let d ∈ DG(x). Apply the Hall-Witt Identity to the elements x, y, and d and obtain the equation

[y, x−1, d]x[x, d−1, y]d[d, y−1, x]y = 1.

Note that [x, d−1, y]d = 1, as [x, d−1] ∈ Z(G). Every commutator centralizes x, and

−1 y −1 −1 so [d, y , x] = 1. It follows that [y, x , d] = 1, and so d ∈ CG([y, x ]). Since

−1 d ∈ DG(x) was arbitrary, DG(x) ≤ CG([y, x ]). Observe the chain

0 −1 −1 Z(G) ≤ G ≤ CG(x) ≤ DG(x) ≤ CG([y, x ]) ≤ DG([y, x ]) ≤ G.

21 Since (G, Z(G)) is a Camina pair and the elements x and [y, x−1] are noncentral,

−1 −1 |DG(x): CG(x)| = |DG([y, x ]) : CG([y, x ])| = |Z(G)|.

The desired inequality follows.

Let (G, Z(G)) be a Camina pair. If G has nilpotence class at least 4 and Z2(G) <

0 CG(G ), then a strong inequality is obtained.

Corollary 4.8. If (G, Z(G)) is a Camina pair, the group G has nilpotence class at

0 least 4, and Z2(G) < CG(G ), then

|G : Z(G)| > |Z(G)|3.

0 2 0 Proof. Because Z2(G) < CG(G ), Theorem 4.7 yields that |G : Z(G)| ≥ |Z(G)| |G : Z(G)|. Now, examine the proof of Theorem 4.3 to see that |G0 : Z(G)| > |Z(G)| when

(G, Z(G)) forms a Camina pair and G has nilpotence class at least 4.

4.4 A Minimal Counterexample to Conjecture (L)

Let (G, Z(G)) be a Camina pair, and suppose that G is a minimal counterexample to Conjecture (L). By the previous results,

• G has nilpotence class 3 (Theorem 4.3),

0 • Z2(G) = CG(G ) (Theorem 4.7), and

• G/Z(G) has exponent p (Lewis, [8], Theorem 5).

But more information can be determined.

Lemma 4.9. Let (G, Z(G)) be a Camina pair, and assume that G is a minimal counterexample to Conjecture (L). Write |Z(G)| = pm. Then |G : Z(G)| = p2m−2.

22 Proof. Let Z ≤ Z(G) with |Z| = p. Note that m > 1 since G is a minimal counterexample to Conjecture (L), and so Z < Z(G). The pair (G/Z, Z(G)/Z) is therefore a Camina pair with Z(G/Z) = Z(G)/Z. Because |G/Z| < |G|, the minimality of G forces |G/Z : Z(G)/Z| ≥ |Z(G/Z)|2 = |Z(G)|2/p2 = p2m−2.

Now, p2m−2 ≤ |G : Z(G)| < |Z(G)|2 = p2m. Hence |G : Z(G)| ∈ { p2m−2, p2m−1 }. By Lemma 2.4 in [8], the index |G : Z(G)| is a square; this fact rules out the possibility

2m−1 that |G : Z(G)| = p .

The following lemma delivers a lower bound for the order of a minimal counterexample.

Lemma 4.10. Let (G, Z(G)) be a Camina pair, and assume that G is a minimal counterexample to Conjecture (L). Write |Z(G)| = pm. Then m ≥ 6, and so |G| ≥ p16.

Proof. Note that m ≥ 5 since 2m − 2 > 3m/2. Suppose that |Z(G)| = p5, and so |G : Z(G)| = p8 by Lemma 4.9. Let x, y ∈ G with [x, y] ∈/ Z(G). Using Lemma 4.4,

5 p ≤ |G : DG(x) ∩ DG(y)| ≤ |G : DG(x)||G : DG(y)|.

5 |DG(x): CG(x)| = p , forcing CG(x) = Z(G), a contradiction.

Recall that if (G, G0) is a Camina pair, then G is called a Camina group. If G is a Camina p-group of nilpotence class 3, then |G : Z(G)| ≥ |Z(G)|2. See various results in [9]. So, any counterexample to Conjecture (L) would not be a Camina group. Note that I.D. MacDonald [11] proved that if G is a class 3 Camina p-group, then (G, Z(G)) is a Camina pair. Thus, if the converse of the previous implication is true, so is the conjecture. Returning to minimal counterexamples to Conjecture (L), it seems natural to examine the smallest counterexample not covered by the previous results; this is our focus for the rest of the section.

23 Notation: a group G satisﬁes (M) if

• (G, Z(G)) is a Camina pair,

• G is a minimal counterexample to Conjecture (L), and

• |G| = p16, where p is an odd prime.

So, those groups that satisfy (M) are the smallest counterexamples to Conjecture (L) not ruled out by the previous results. Note that if G satisﬁes (M), then |Z(G)| = p6 and |G : Z(G)| = p10. As mentioned in Chapter 1, Lewis established Conjecture (L) in the situation where the factor group G/Z(G) has exponent at least p2. Hence, if G satisﬁes (M), then the exponent of G/Z(G) is p. If p = 2, then, using the fact that groups of exponent 2 are abelian, G has class 2, which is not the case. Thus, p must be an odd prime, explaining the third bulleted item.

Lemma 4.11. If G satisﬁes (M), then

3 1. |G : DG(x)| = p for every x ∈ G \ Z2(G), and

2. |G0 : Z(G)| ∈ {p3, p4}.

4 Proof. Let x ∈ G \ Z2(G). If |G : DG(x)| ≥ p , then

10 4 6 11 p = |G : Z(G)| = |G : DG(x)||DG(x): CG(x)||CG(x): Z(G)| ≥ p p p = p ,

2 a contradiction. Assume that |G : DG(x)| ≤ p . Choose y ∈ G \ DG(x). Note that

y∈ / Z2(G). By Lemma 4.4,

6 2 p = |Z(G)| ≤ |G : DG(x) ∩ DG(y)| ≤ |G : DG(x)||G : DG(y)| ≤ p |G : DG(y)|.

4 But now, |G : DG(y)| ≥ p , contrary to the fact just proved. Item (1) has been established.

24 0 6 0 4 By Lemma 3.11, |G : G | ≥ p . Hence |G : Z(G)| ≤ p . Let x ∈ G \ Z2(G) and observe that

0 3 |G : Z(G)| ≥ |G : CG(x)|/|Z(G)| = |G : DG(x)| = p .

0 3 4 Indeed, |G : Z(G)| ∈ {p , p }.

The next lemma considers the factor group G/Z(G) for a group G satisfying (M).

Lemma 4.12. If G satisﬁes (M) and n is the size of a conjugacy class in G/Z(G), then n ∈ {1, p3}.

Proof. Of course, the size of the conjugacy class of a central element in G/Z(G) is 1.

If Z(G)x∈ / Z(G/Z(G)) = Z2(G)/Z(G), then x ∈ G \ Z2(G). Thus,

G/Z(G) 3 |(Z(G)x) | = |G/Z(G): CG/Z(G)(Z(G)x)| = |G : DG(x)| = p ,

using the previous lemma. The result follows.

Tushar Kanta Naik and Manoj K. Yadav [13] classiﬁed, up to isoclinism, the ﬁnite p-groups that have conjugacy class sizes of 1 and p3. Lemma 3.2 in [13] is particularly relevant here.

Lemma (Naik and Yadav, [13], Lemma 3.2). Let G be a ﬁnite p-group, and assume that G has conjugacy class sizes of 1 and p3. Then one of the following situations occurs:

1. |G0| = p3 and |G : Z(G)| ≥ p4, or

2. |G0| ≥ p4 and |G : Z(G)| = p4.

The preceding lemma narrows down the structure of groups G satisfying (M). In particular, applying this lemma to the factor group G/Z(G) yields precise information about |G0 : Z(G)|.

25 Lemma 4.13. If G satisﬁes (M), then |G0 : Z(G)| = p3.

Proof. The factor group G/Z(G) satisﬁes the hypotheses of the previous lemma. Note that

6 |G/Z(G): Z(G/Z(G))| = |G : Z2(G)| ≥ p .

3 0 0 The previous lemma now yields that p = |(G/Z(G)) | = |G : Z(G)|.

0 If G satisﬁes (M), then |Z2(G): G | ∈ {1, p}. But one of these situations can be ruled out.

0 Lemma 4.14. If G satisﬁes (M), then |Z2(G): G | = p.

0 0 G 9 0 Proof. Assume that Z2(G) = G . If x ∈ G \ G , then |x | = |G : CG(x)| = p = |G |. Therefore G is a Camina p-group of nilpotence class 3, and so |G : Z(G)| ≥ |Z(G)|2, a

contradiction.

4.5 More Results on Central Camina Pairs

A variety of results concerning central Camina pairs is presented in this section. First, if (G, Z(G)) is a Camina pair, then a certain factorization of Z(G) leads to an alternate proof of Theorem 4.5. Then, a bound obtained in the proof of Theorem 4.3 is extracted. This bound allows us to add an additional condition to a potential minimal counterexample to Conjecture (L).

Lemma 4.15. Let (G, Z(G)) be a Camina pair, and let Z < Z(G). If C is the

0 0 subgroup of G such that C/Z = CG/Z ((G/Z) ), then C ≤ Z(G).

Proof. Observe that [C, G, C] = [G, C, C] ≤ [G0,C] ≤ Z. By the Three Subgroups Lemma, [C0,G] = [C,C,G] ≤ Z. It follows that C0Z/Z ≤ Z(G/Z) = Z(G)/Z, where

the equality is a consequence of Lemma 3.9.

Next, a factorization of the center is obtained.

26 Lemma 4.16. Let (G, Z(G)) be a Camina pair and assume that G has nilpotence class 3. If x, y ∈ G with [x, y] ∈/ Z(G), then Z(G) = [G0, x][G0, y].

Proof. Since G has nilpotence class 3, the subgroups [G0, x] and [G0, y] are contained in Z(G). Suppose that [G0, x][G0, y] < Z(G). Write Z = [G0, x][G0, y], and let C be

0 the subgroup of G such that C/Z = CG/Z ((G/Z) ). The elements x and y centralize G0 modulo Z; in other words, x, y ∈ C. But, by Lemma 4.15, C0 ≤ Z(G), against the

choice of x and y.

The previous factorization of Z(G) provides an alternate proof of Theorem 4.5.

Alternate Proof of Theorem 4.5. Let x, y ∈ G with [x, y] ∈/ Z(G). Note that Z(G) = [G0, x][G0, y] by Lemma 4.16. By Lemma 3.5, the group G is a p-group for some prime

m 0 i(x) 0 i(y) p. Write |Z(G)| = p , |G : CG0 (x)| = |[G , x]| = p , |G : CG0 (y)| = |[G , y]| = p , and |[G0, x] ∩ [G0, y]| = pd for nonnegative integers m, i(x), i(y), and d. Observe that

pm = |Z(G)| = |[G0, x][G0, y]| = pi(x)+i(y)−d,

and so m = i(x) + i(y) − d. If d 6= 0, then i(x) + i(y) = m + d > m. If i(x), i(y) ≤ m/2, then m < i(x) + i(y) ≤ m, a contradiction. Hence, i(x) > m/2 or i(y) > m/2. Without loss of generality, assume that i(x) > m/2. As |G : G0| ≥ |Z(G)| = pm,

0 0 |G : Z(G)| = |G : G ||G : CG0 (x)||CG0 (x): Z(G)|

≥ pmpi(x) > pmpm/2 = p3m/2 = |Z(G)|3/2.

The inequality |G : Z(G)| > |Z(G)|3/2 therefore holds in this case. If d = 0, then i(x) + i(y) = m. If either i(x) > m/2 or i(y) > m/2, then argue as in the previous case. Assume that i(x), i(y) ≤ m/2. This situation forces i(x) = i(y) = m/2. Work with i(x) and use an inequality from Theorem 4.3: namely,

0 m m/2 m/2 |G : DG(x)| ≥ |Z(G):[G , x]| = p /p = p .

27 Now,

|G : Z(G)| = |G : DG(x)||DG(x): CG(x)||CG(x): Z(G)|

m/2 m ≥ p p |CG(x): Z(G)|

> pm/2pm = p3m/2 = |Z(G)|3/2.

3/2 The inequality |G : Z(G)| > |Z(G)| holds in this case, too.

A particular bound that was obtained in the proof of Theorem 4.3 is now extracted here as technical lemma.

Lemma 4.17. If (G, Z(G)) is a Camina pair where G has nilpotence class at least 3 and x ∈ Z3(G) \ Z2(G), then

|Z(G)| |G : DG(x)| ≥ 0 . |G : CG0 (x)|

0 0 Proof. By Lemma 4.2, [G , x] is a central subgroup of order |G : CG0 (x)|. Write Z = [G0, x]. If Z = Z(G), then the quantity on the right-hand side of the inequality in the conclusion of this lemma is 1. So this case is trivial. Assume that Z < Z(G), and note that (G/Z, Z(G)/Z) is Camina pair with

−1 Z(G)/Z = Z(G/Z). Choose y ∈ G \ DG(x). The commutator [y, x ] is noncentral and, as Z(G/Z) = Z(G)/Z, the commutator [Zy, Zx−1] = Z[y, x−1] is also noncen-

−1 tral. Using the Correspondence Theorem, write C/Z = CG/Z (Z[y, x ]). Because (G/Z, Z(G)/Z) is a Camina pair, it follows that |G/Z : C/Z| ≥ |Z(G/Z)| = |Z(G)|/|Z|.

Let d ∈ DG(x). Applying the Hall-Witt Identity to the elements d, x, and y,

[y, x−1, d]x[x, d−1, y]d[d, y−1, x]y = 1.

Note that [x, d−1, y]d = 1 as [x, d−1] ∈ Z(G). Hence,

[y, x−1, d] ∈ (([d, y−1, x]y)−1)x−1 ∈ Z.

28 Now, as d ∈ DG(x) was arbitrary, DG(x) ≤ C. Hence,

|G : DG(x)| ≥ |G : C| ≥ |Z(G)|/|Z|.

0 As |Z| = |G : CG0 (x)|, the desired inequality follows.

By the previous lemma, a minimal counterexample to Conjecture (L) must satisfy another condition.

Lemma 4.18. Let (G, Z(G)) be a Camina pair. If G is a minimal counterexample to

0 Conjecture (L), then |CG(x)| < |G | for each x ∈ G \ Z2(G).

Proof. Let x ∈ G \ Z2(G). By Lemma 4.17,

|Z(G)| |G : DG(x)| ≥ 0 . |G : CG0 (x)|

Note that |DG(x): CG(x)| = |Z(G)| since x∈ / Z(G). It follows that

2 |CG(x): Z(G)| |G : Z(G)| ≥ |Z(G)| 0 . |G : CG0 (x)|

0 0 Now, since |G : CG0 (x)| ≤ |G : Z(G)|,

1 1 0 ≥ 0 . |G : CG0 (x)| |G : Z(G)|

Hence, |C (x): Z(G)| |G : Z(G)| ≥ |Z(G)|2 G . |G0 : Z(G)| Using the hypothesis,

|C (x): Z(G)| |Z(G)|2 > |Z(G)|2 G . |G0 : Z(G)|

The result follows.

29 CHAPTER 5

A PARTICULAR GRAPH ATTACHED TO A GROUP

The topic in this chapter is independent of the previous results on Camina pairs and concerns a particular graph attached to a group.

5.1 Introduction

Let G be a group. Take G# = G \{ 1 } as a vertex set and draw an edge between distinct elements x and y of G# if and only if hx, yi is cyclic. The graph just described is called the cyclic graph of G and is denoted by ∆(G). If the connected components of ∆(G) are complete graphs, then G is called cyclic transitive. Diana Imperatore [5] imposed cyclic-transitivity on various classes of groups and obtained strong classification results. In a joint paper by D. Imperatore and M.L. Lewis [6], the finite cyclic-transitive groups were classified. The focus in this chapter is to characterize the cyclic graph of a group under various group-theoretic assumptions. In particular, the primary concern is describing ∆(G) for a 2-Frobenius group G. A group G is a 2-Frobenius group if it has normal subgroups K and L such that L is a Frobenius group with Frobenius kernel K and G/K is a Frobenius group with Frobenius kernel L/K. It is easy to check that ∆(G) is disconnected for a 2-Frobenius group G, and so our goal is to count the number of connected components. This task turned out to be difficult, but here are some results in this direction.

Theorem F. Let G be a 2-Frobenius group with K as in the deﬁnition, and let D be a subgroup of G such that D/K is a Frobenius complement of the factor group G/K. If |K| is divisible by at least two distinct prime numbers, then ∆(G) has |K| + 1

30 connected components.

The hypothesis that K is not a group of prime power order is now dropped. But, the subgroup D is now assumed to be a p-group for some prime number p. Note that

D ∈ Sylp(G) under these conditions. We introduce some notation. If G is a group and

p is a prime number, then write mp(G) for the number of subgroups of order p of G

Theorem G. Let G be a 2-Frobenius group, and assume that D is a p-group for some prime number p, where D is as described above. Then ∆(G) has |K| + mp(G) connected components.

Before getting started, however, a few more elementary group-theoretic results need to be collected, and the prerequisite concepts from graph theory need to be formally introduced. Related graphs are also discussed.

5.2 More Elementary Group Theory

Lemma 5.1 (Frattini Argument). Let G be a group acting transitively on a set Ω, H be a subgroup of G, and α be any point in Ω. Then H acts transitively on Ω if and only if G = GαH.

Lemma 5.2 (N/C-Theorem). If H ≤ G, then NG(H)/CG(H) is isomorphic to a subgroup of Aut(H).

The next omnibus theorem gathers together many basic facts about Frobenius groups.

Theorem 5.3. Let G be a Frobenius group with Frobenius kernel K and Frobenius complement H.

1. The subgroup K is nilpotent.

2. If N is a normal subgroup of G, then N ≤ K or K ≤ N.

31 3. The Sylow subgroups of H are cyclic or generalized quaternion.

4. If |H| is even, then H has a unique involution.

5. If |H| is odd, then H has a unique subgroup of order p for every prime divisor p of |H|.

# 6. If h ∈ H , then CG(h) ≤ H.

# 7. If k ∈ K , then CG(k) ≤ K.

As mentioned, the facts in Theorem 5.3 are basic; but here are some references. Item (1) is a consequence of John G. Thompson’s Ph.D. thesis. Item (2) appears as Theorem 12.6.8 in [15]. Item (3) is Theorem 12.6.15 in [15]. Item (4) is a part of Theorem 10.3.1 in [4]. Item (5) is Theorem 6.19 in [7]. The remaining items appear in all of the references just mentioned.

Lemma 5.4. If H is a Frobenius complement and p is the smallest prime divisor of |H|, then H has a unique subgroup of order p and this subgroup is central.

Proof. If |H| is even, then H has a unique involution, which will necessarily be central. If |H| is odd, then H has a unique subgroup of order q for every prime divisor q of

|H|. Let P be the unique subgroup of order p. By the N/C-Theorem, |H : CH (P )|

divides p − 1. A prime divisor of |H : CH (P )| would divide p − 1, and so the

hypothesis that p is the smallest prime divisor of |H| therefore forces H = CH (P ).

Hence P ≤ Z(H).

5.3 Basic Graph Theory

A graph is an ordered pair (V,E) consisting of a ﬁnite set V and a set E ⊆ V × V such that

• (v, w) ∈ E if and only if (w, v) ∈ E

32 • (v, v) ∈/ E for each v ∈ V .

The elements of V are called vertices. The elements of E are called edges. So, diagrammatically, the graphs under consideration are undirected and have no loops and no multiple edges. If (v, w) ∈ E, then the vertex v is adjacent to w. A symbol such as ∼ or ≈ shall be used to denote adjacency in this dissertation. Note that adjacency is an irreﬂexive, symmetric relation. Let Γ be a graph and denote adjacency by ∼.A path of length n in Γ is a collection

of vertices v0, . . . , vi, . . . vn such that

v0 ∼ · · · ∼ vi ∼ · · · ∼ vn.

If v and w are vertices in Γ and there exists at least one path between v and w, then d(v, w) denotes the length of the shortest path between them. The integer d(v, w) is called the distance between v and w. If there does not exist a path between v and w, then d(v, w) = ∞. Also, for each vertex v in Γ, write d(v, v) = 0. Again, let Γ be a graph. Deﬁne a relation ≡ on Γ by writing v ≡ w, for vertices v and w, if and only if d(v, w) < ∞. The relation ≡ is easily seen to be an equivalence relation on Γ, and the equivalence classes under this relation are called connected components. If Γ has just one connected component (in other words, there exists a path between any two vertices), then Γ is said to be connected; otherwise, Γ is disconnected. If Γ is connected, then diam(Γ) = max{ d(u, v) | u, v ∈ V }.

5.4 Preliminary Observations

Let G be a group. As constructed in the Introduction, deﬁne a graph ∆(G) by letting the set G# = G \{ 1 } be the set of vertices and by declaring two distinct elements x, y ∈ G to be adjacent if and only if the subgroup hx, yi is cyclic. The graph ∆(G) is called the cyclic graph of G. For x, y ∈ G#, the expression x ≈ y denotes adjacency in ∆(G).

33 Fix a group G. The following graphs are related to the cyclic graph. Let X ⊆ G. The power graph of G with respect to X is constructed by declaring distinct elements x, y ∈ X to be adjacent if and only if x ∈ hyi or y ∈ hxi. The commuting graph of G with respect to X is constructed by declaring distinct elements x, y ∈ X to be adjacent if and only if xy = yx. Usually, X ∈ {G#,G} when the power graph is under consideration and X ∈ {G \ Z(G),G#,G} when the commuting graph is under consideration. For the remainder of this chapter, let Γ(G) denote the commuting graph of G with respect to X = G \ Z(G). Write ∼ for adjacency in Γ(G): so x ∼ y if and only if xy = yx for x, y ∈ X. Returning to the cyclic graph, there is an easy suﬃcient condition for adjacency in ∆(G).

Lemma 5.5. Let G be a group, and let x, y ∈ G. If x and y are commuting elements with coprime orders, then the subgroup hx, yi is cyclic.

The condition that elements with coprime orders commute characterizes ﬁnite nilpotent groups. (See Theorem 2.12 in [16].) With this fact in mind, ∆(G) can be characterized for a nilpotent group G that is not a group of prime power order.

Lemma 5.6. If G is nilpotent and there exist at least two distinct primes that divide |G|, then ∆(G) is connected with diam(∆(G)) ≤ 3.

Proof. Let x, y ∈ G#. Suppose that there exists a prime p that divides o(x) and does not divide o(y). Note that o(xt) = p for suitable integer t, and so x ≈ xt ≈ y. Assume that every prime divisor of o(x) is a prime divisor of o(y). Also, assume that every prime divisor of o(y) is a prime divisor of o(x). (Otherwise, apply the argument in the previous paragraph.) If o(x) and o(y) are powers of some prime p, then, using the hypothesis, ﬁx a ∈ G# with o(a) = q, where q 6= p is a prime. Observe that x ≈ a ≈ y. If o(x) and o(y) are not powers of some prime, then there exist

34 suitable integers r and s and distinct primes p and q such that o(xt) = p and o(ys) = q.

t s Now, x ≈ x ≈ y ≈ y.

The cyclic graph of a group whose center is not a group of prime power order is easily described.

Lemma 5.7. If G is a group and there exist at least two distinct primes that divide |Z(G)|, then ∆(G) is connected with diam(∆(G)) ≤ 4.

Proof. Let p and q be distinct prime divisors of |Z(G)|. Let a, b ∈ Z(G) with o(a) = p and o(b) = q. Note that a ≈ b. Let x ∈ G#. If a prime r 6= p divides o(x), then o(xt) = r for a suitable integer t. Now, x ≈ xt ≈ a. If x is a p-element, then x ≈ b, and so x ≈ b ≈ a. Hence d(x, a) ≤ 2

# for every x ∈ G , and the result follows.

Next, let G be a Frobenius group. It turns out that ∆(G) is easily understood.

Lemma 5.8. If G is a Frobenius group with Frobenius complement H, then H# is a connected component of ∆(G).

Proof. Let p be the smallest prime divisor of |H|, and let x ∈ H# with o(x) = p. By Lemma 5.4, hxi is the unique subgroup of H with order p and hxi ≤ Z(H). Let h ∈ H#. If p does not divide o(h), then h ≈ x by Lemma 5.5. If p divides o(h), then ht = x for some integer t; hence h ≈ x.

# # Finally, note that if g ∈ G with g ≈ h for some h ∈ H , then g ∈ CG(h) ≤ H.

# It follows that H constitutes a connected component of ∆(G).

Notice that if H is a Frobenius complement, then diam(∆(H)) ≤ 2. If G is a group, then write ν(∆(G)) for the number of connected components of ∆(G).

Theorem 5.9. If G is a Frobenius group with Frobenius kernel K and Frobenius complement H, then ∆(G) has |K| + ν(∆(K)) connected components.

35 Proof. There are |G : H| = |K| Frobenius complements of G, and each complement constitutes a connected component of ∆(G) by Lemma 5.8. Since

[ K# = G \ Hg, g∈G the points in G# that do not belong to a Frobenius complement are exactly the points in K#. Hence the remaining connected components of ∆(G) are precisely the

connected components of ∆(K).

Let G be a Frobenius group with Frobenius kernel K, and consider the commuting graph Γ(G). The preceding results yield a count on the number of connected components of Γ(G). In particular, Γ(G) is disconnected with |K| + 1 connected components.

5.5 2-Frobenius Groups

Now, results from our investigation of ∆(G) for a 2-Frobenius group G are presented. Recall that a group G is called a 2-Frobenius group if there exist normal subgroups K and L of G such that L is a Frobenius group with Frobenius kernel K and G/K is a Frobenius group with Frobenius kernel L/K. The following notation will remain in eﬀect for the remainder of the section. A group G is said to satisfy (F) if

• G is a 2-Frobenius group with normal subgroups K and L as in the deﬁnition,

• H is a Frobenius complement of L,

• N = NG(H), and

• D is a subgroup of G such that D/K is a Frobenius complement of G/K.

The symmetric group on four letters, S4, is an example of a 2-Frobenius group.

The unique minimal normal subgroup of S4, which is a Klein 4-group, plays the role

of K, and A4, the alternating group on four letters, plays the role of L.

36 Structural properties of 2-Frobenius groups are alluded to at various places in the literature. (For example, some of the facts below are used implicitly in [14].) For the sake of completeness, here are a few basic results.

Lemma 5.10. If G satisﬁes (F), then N is a Frobenius group with Frobenius kernel H.

Proof. By a Frattini Argument, G = KN. Note that G/K ∼= N as K ∩ N =

NK (H) = { 1 }, and so N is a Frobenius group. The Frobenius kernel of N has order |L : K| = |H|. Since H is normal in N, the Frobenius kernel of N is H by Item (2)

in Theorem 5.3.

Since the subgroup H is simultaneously a Frobenius kernel and a Frobenius complement, its structure is quite limited.

Lemma 5.11. If G satisﬁes (F), then H is cyclic of odd order.

Proof. Seeking a contradiction, assume that |H| is even. Therefore, as H is a Frobenius complement, H has a unique involution; call it z. Note that o(z) = o(zσ) for every

σ ∈ Aut(H). Hence z = zσ, and so z ∈ FixH# (Aut(H)). In particular, H does not admit a ﬁxed-point-free automorphism. But H is also a Frobenius kernel. These conditions are not compatible; thus |H| must be odd. As H is a Frobenius kernel, every Sylow subgroup of H is normal in H. As H is a Frobenius complement of odd order, the Sylow subgroups of H are cyclic. Thus H is

cyclic.

The structure of a Frobenius complement in N is also easily determined.

Lemma 5.12. If G satisﬁes (F), then ND(H) is cyclic.

Proof. Note that G = HD, and so

N = N ∩ HD = H(N ∩ D) = HND(H),

37 by Dedekind’s Lemma. As H ∩ ND(H) = { 1 }, the subgroup ND(H) is a Frobenius

complement in N. Observe that CD(H) = { 1 }. Thus ND(H) embeds in Aut(H). By

Lemma 5.11, Aut(H) is abelian. Hence ND(H) is an abelian Frobenius complement, which yields the conclusion.

Lemma 5.13. If G satisﬁes (F), then D = KND(H).

Proof. Observe that

D = D ∩ G = D ∩ KN = K(D ∩ N) = KND(H),

by Dedekind’s Lemma.

Let G be a group. Recall that if H is a subgroup of G such that (|H|, |G : H|) = 1, then H is a Hall subgroup of G.

Lemma 5.14. If G satisﬁes (F), then D is a Hall subgroup of G.

Proof. Notice that |G : D| = |H|. Using Lemma 5.13, |D| = |K||ND(H)|. The result

follows.

Results concerning the cyclic graph of a 2-Frobenius group are now recorded.

Lemma 5.15. If G satisﬁes (F) and g ∈ G# with g ≈ d for some d ∈ D \ K, then g ∈ D.

Proof. Since Kd ∈ (D/K)#,

Kg ∈ CG/K (Kd) ≤ D/K.

Hence g ∈ D.

Suppose that G satisﬁes (F), and let X ∈ DG \{ D }. As a consequence of Lemma 5.15, the only way to build a path between points in D \ K and X \ K is through K.

Lemma 5.16. If G satisﬁes (F), then H# is a connected component of ∆(G).

38 # Proof. If h1, h2 ∈ H , then, by Lemma 5.8, there exists a path between h1 and h2 through H#. Let h ∈ H#, and suppose that g ∈ G# with g ≈ h. Write g = ak, where a ∈ N, k ∈ K. Observe that h ∈ H ∩ Hak = H ∩ Hk since h = hg = hak. Now

k H = H , and so k ∈ K ∩ H = { 1 }. Hence g = a ∈ CN (h) ≤ H.

Lemma 5.16 holds for each conjugate of H#. Hence the set of conjugates of H# constitutes |K| connected components of ∆(G). In fact, Lemma 5.16 shows that each conjugate of H# is a connected component of Γ(G), the commuting graph of G with respect to G \ Z(G) = G#. (A 2-Frobenius group has a trivial center.) After a technical lemma, the number of connected components of Γ(G) is determined.

S g Lemma 5.17. Let G satisfy (F), and let x ∈ G \ g∈G H . If o(x) = p, a prime, then CK (x) 6= { 1 }.

Proof. Conjugation by x induces an automorphism of L. If CL(x) = { 1 }, then L admits a ﬁxed-point-free automorphism of prime order and is therefore nilpotent, a

contradiction. Hence CL(x) 6= { 1 }. By Lemma 5.16, x must centralize an element in

S g # the set L \ g∈L H = K .

S y # Let G satisfy (F), and let g ∈ G \ y∈G H and fix z ∈ Z(K) . Raise g to an appropriate power in order to obtain an element x of prime order. Let k ∈ K# be an element that centralizes x, as delivered by the previous lemma. Hence g ∼ x ∼ k ∼ z. It follows that Γ(G) has |K| + 1 connected components. Obtaining a count on the number of connected component of ∆(G), on the other hand, seems more difficult. But some work in this direction is now presented. As a slight help, another condition is introduced. A group G is said to satisfy (F 0) if it satisfies (F) and |K| is divisible by at least two distinct primes.

Theorem 5.18. If G satisﬁes (F 0), then ∆(G) has |K| + 1 connected components.

# S g Proof. By Lemma 5.16, the set g∈G H is partitioned into |K| connected components of ∆(G).

39 S g Let d ∈ G \ g∈G H . The task is to show that there exists a path from d into K#. Raise d to an appropriate power to obtain an element x of prime order p. If x ∈ K, then d ≈ x is a path from d into K#. So, assume that x∈ / K. Let

Q ∈ Sylq(K), where q 6= p, and let K0 be the normal q-complement of K. Write

G = G/K0, and note that G is a 2-Frobenius group. By Lemma 5.17, x centralizes # an element y ∈ K . Hence [x, y] ∈ K0. The coset representative y can be chosen

to belong to Q. Note that [x, y] ∈ Q as Q E G. Hence [x, y] ∈ Q ∩ K0 = { 1 }. Because x and y commute and have coprime orders, x ≈ y. Now, d ≈ x ≈ y. As

S g |K| is divisible by at least two distinct primes, the set G \ g∈H H constitutes a

connected component of ∆(G).

One more count of the number of connected components of ∆(G) for a 2-Frobenius group G is oﬀered. This time, the condition that |K| is not a prime power is dropped. But, the assumption that the subgroup D has prime power order is added. An easy preliminary observation needs to be mentioned.

Lemma 5.19. If X is p-group for some prime number p, then the number of connected components of ∆(X) is equal to the number of subgroups of order p.

Recall from the Introduction that if G is a group and p is a prime number, then

mp(G) denotes the number of subgroups of order p in G.

Theorem 5.20. If G satisﬁes (F) and D is a p-group for some prime p, then ∆(G) has |K| + mp(G) connected components.

# S g Proof. The set g∈G H is partitioned into |K| connected components of ∆(G). Note that !# [ [ G \ Hg = Dg . g∈G g∈G

The hypothesis and Lemma 5.14 yield that D ∈ Sylp(G). So all of the subgroups of order p of G are contained in conjugates of D. Lemma 5.15 ensures that no point in # S g the set G \ g∈G D is adjacent to a point in any conjugate of D.

40 So, we are left to show that if hxi and hyi are distinct subgroups of G of order p, then x and y belong to distinct connected components of ∆(G). For a contradiction, suppose that x and y lie in the same connected component of ∆(G), say Ξ. Since

S g Ξ ⊆ g∈G D , every element of Ξ is a p-element. Now, write d(x, y) = n, and note that n > 1. Let

x ≈ x1 ≈ x2 ≈ · · · ≈ xn−1 ≈ y

be a path of length n. The subgroup hx, x1i has a unique subgroup of order p. Since x1

t t is a p-element, o(x1) = p for some positive integer t. Thus, hxi = hx1i. Similarly, the

s subgroup hx1, x2i has a unique subgroup of order p. As x2 is a p-element, o(x2) = p

t s s u for some positive integer s. Hence hx1i = hx2i. But now hxi = hx2i, and so x = x2

for some positive integer u. It follows that x ≈ x2, which implies d(x, y) ≤ n − 1, a

contradiction.

A slightly more explicit way to frame the count on the number of connected

components in Theorem 5.20 is |K| + |G : D|(mp(D) − mp(K)) + mp(K). The previous results leave unanswered the question of the number of connected components of ∆(G) when |K| is a prime power and |D| is divisible by at least two distinct prime numbers. A much more detailed analysis seems to be necessary in this case. Assuming that |K| is a power of p, where p is a prime number, p0-elements in D are adjacent to elements in K#. Controlling how distinct connected components of ∆(K) are fused by these p0-elements and, of course, understanding the global structure of ∆(D) will need more work.

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