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P-GROUP CODEGREE SETS and NILPOTENCE CLASS A

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P-GROUP CODEGREE SETS and NILPOTENCE CLASS A p-GROUP CODEGREE SETS AND NILPOTENCE CLASS A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Sarah B. Croome May, 2019 Dissertation written by Sarah B. Croome B.A., University of South Florida, 2013 M.A., Kent State University, 2015 Ph.D., Kent State University, 2019 Approved by Mark L. Lewis , Chair, Doctoral Dissertation Committee Stephen M. Gagola, Jr. , Members, Doctoral Dissertation Committee Donald L. White Robert A. Walker Joanne C. Caniglia Accepted by Andrew M. Tonge , Chair, Department of Mathematical Sciences James L. Blank , Dean, College of Arts and Sciences TABLE OF CONTENTS Table of Contents . iii Acknowledgments . iv 1 Introduction . 1 2 Background . 6 3 Codegrees of Maximal Class p-groups . 19 4 Inclusion of p2 as a Codegree . 28 5 p-groups with Exactly Four Codegrees . 38 Concluding Remarks . 55 References . 55 iii Acknowledgments I would like to thank my advisor, Dr. Mark Lewis, for his assistance and guidance. I would also like to thank my parents for their support throughout the many years of my education. Thanks to all of my friends for their patience. iv CHAPTER 1 Introduction The degrees of the irreducible characters of a finite group G, denoted cd(G), have often been studied for their insight into the structure of groups. All groups in this dissertation are finite p-groups where p is a prime, and for such groups, the degrees of the irreducible characters are always powers of p. Any collection of p-powers that includes 1 can occur as the set of irreducible character degrees for some group [11]. In fact, the group can always be chosen so that its nilpotence class is at most 2. The question of when a set of character degrees bounds the nilpotence class is also of interest, and the answer is hardly straightforward. Even for groups with just two character degrees, there are starkly contrasting possibilities. In [13], it is shown that f1; pg can occur as the character degree set of groups with arbitrarily large class, while Theorem 3.10 of [14] states that if cd(G) = f1; peg for e > 1, then the class of G is at most p. The purpose of this dissertation is to investigate the set of codegrees of a group and its relationship to nilpotence class. The codegree of an irreducible character χ of a finite group G is defined by cod(χ) = jG : ker(χ)j/χ(1), the index of the kernel of χ in G, divided by the degree of χ. The set of codegrees of the irreducible characters of G is denoted cod(G). This definition for codegrees first appeared in [17] in 2007, where the authors use a graph-theoretic approach to compare the structure of a group with its set of codegrees. Earlier use of the term codegree appears in [6] in 1989, with the slightly different definition jGj/χ(1). In their paper, Chillag and Herzog examine the structure of groups whose codegrees are divisible by at most two primes. This definition, without the actual term codegree, appeared even earlier 1 in [9] in 1981. In his paper, Gagola showed that for a finite group G with normal subgroup N, if χ 2 Irr(N) is invariant in G, then χ is extendible to G when (jG : Nj; jNj/χ(1)) = 1. Returning to the modern defintion of codegrees, in 2016, Du and Lewis proved in [7] that when G is a finite p-group and jcod(G)j = 3, the nilpotence class of G, denoted c(G), is at most 2, suggesting that groups with few codegrees may have bounded nilpotence class. Our original purpose was to find a sharp bound for the nilpotence class of groups with exactly four codegrees. When a group has exactly 4 codegrees and some additional restrictions we can show that the nilpotence class of the group is at most 4. Theorem 1. Let G be a finite p-group such that cod(G) = f1; p; pb; pag, where 2 ≤ b < a. If any of the following hold, then G has nilpotence class at most 4: (i) jcd(G)j = 2, (ii) cd(G) = f1; p; p2g, (iii) jG : G0j = p2. The coclass of a p-group G with nilpotence class c is given by logp jGj − c. If G has four codegrees and coclass at most 3, then G has nilpotence class at most 4, bounding the order of G for a given prime p. Theorem 2. Let G be a p-group such that cod(G) = f1; p; pb; pag, where 2 ≤ b < a. If G has coclass at most 3, then G has nilpotence class at most 4, and jGj ≤ p7. With the following additional hypothesis, we can extend the result of Theorem 2 to p-groups with coclass at most 7. 2n Hypothesis (∗). If G is a p-group with nilpotence class n such that jGj ≥ p , then jZ2(G)j 6= p2. 2 Theorem 3. Let G be a finite p-group with cod(G) = f1; p; pb; pag, where 2 ≤ b < a. If G has coclass at most 7, and G and all of its quotients satisfy Hypothesis (∗), then the nilpotence class of G is at most 4 and jGj ≤ p11. In [7], Du and Lewis were able to bound c(G) in terms of the largest member of cod(G). They showed that if pa, (where a > 1), is the largest codegree of G, then c(G) ≤ 2a − 2, and in some specific cases, c(G) ≤ 2a − 3. When jcod(G)j = 4, we can improve this bound. Theorem 4. If G is a finite p-group with cod(G) = f1; p; pb; pag, where 2 ≤ b < a, then c(G) ≤ a + 1. This bound can be improved slightly when the two largest codegrees are consecutive powers of p and the group does not have p2 as a codegree. Notice that in Theorem 5, p2 2= cod(G). The question of when p2 must be included in the set of codegrees of a group motivates the results in chapter 4. Theorem 5. If G is a finite p-group such that cod(G) = f1; p; pa−1; pag for a ≥ 4, then c(G) ≤ a. It is also interesting to ask about the nilpotence class of G when cod(G) contains as many codegrees as possible for the group's order. Surprisingly, if G has order pn and cod(G) contains every power of p up to pn−1, it turns out that the nilpotence class of G can either be quite small or as large as possible, but not in between. This is the result of Theorem 6. Theorem 6. If G is a p-group such that jGj = pn ≥ p2, then cod(G) = f1; p; p2; : : : ; pn−1g if and only if one of the following occurs: ∼ (i) G = Zpn−1 × Zp, ∼ (ii) G = Zpn−1 o Zp and has nilpotence class 2, (iii) G has maximal class and jcd(G)j = 2. 3 While examining maximal class p-groups, it became apparent that the codegrees of these groups are often consecutive powers of p. From Theorem 6, we know that a maximal class group with order pn+1 will only have all powers of p up to pn as codegrees if the group has exactly two character degrees. With three character degrees, the codegrees are still consecutive powers of p, but the largest power may vary. Theorem 7. Let G be a maximal class p-group for some prime p such that cd(G) = f1; p; pbg. If jGj = pn, then cod(G) = fpi j 0 ≤ i ≤ cg, for some integer n − b ≤ c ≤ n − 2. Any metabelian maximal class group can have at most three character degrees, so we know from Theorem 7 that the codegrees of such a group will be consecutive powers of p. In this case, however, there are only two possibilities for the largest power. Corollary 8. Let G be a metabelian maximal class p-group for some prime p. If jGj = pn, then cod(G) = fpi j 0 ≤ i ≤ cg, where c = n − 1 or n − 2. The codegrees of a normally monomial maximal class p-group will also be consecutive p-powers. This result is similar to that of Theorem 7, but does not depend on the number of character degrees of the group. Theorem 9. Let G be a normally monomial maximal class p-group for some prime p. Let n i jGj = p , b(G) = maxfχ(1) j χ 2 Irr(G)g, and b = logp(b(G)). Then cod(G) = fp j 0 ≤ i ≤ cg for some integer c ≥ n − b. It is easily shown that p2 and p3 are always included among the codegrees of maximal class p-groups (when the order of the group is large enough for the codegree to occur, e.g., p3 2 cod(G) when jGj ≥ p4). If the order of G is at least p6, G also has p4 as a codegree. This evidence, including Theorems 6, 7, 9, and Corollary 8, leads us to ask whether the codegrees of maximal class p-groups are always consecutive powers of p. 4 In the case of character degrees of p-groups, there is one (non-trivial) degree that stands out as more influential than the rest. In [15], it is shown that sets of p-power character degrees that include p are not class bounding. The set of codegrees of a p-group always includes p [7, Corollary 2.3], but the next smallest power, p2, plays an important role when determining nilpotence class.
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