CHARACTER TABLES OF FINITE GROUPS By ADRIANA NENCIU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Adriana Nenciu ACKNOWLEDGMENTS My ¯rst and foremost thanks go to my advisor, Dr. Alexandre Turull. Without his constant guidance and support my doctoral research would never have been possible. The members of my doctoral dissertation committee, Dr. Ion Giviriga, Dr. Jorge Martinez, Dr. Peter Sin, and Dr. Pham Tiep, have been very generous with their time and help. I am grateful to all the mathematics teachers I have had, especially to Ioana Andrei and Dr. Victor Vuletescu. I wish to thank my friends from Bucharest for their interest in my work and for our discussions on mathematical and non-mathematical topics. Last but not least I want to thank Zia for his patience, understanding, and support. iii TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................. iii TABLE ....................................... vi ABSTRACT .................................... vii CHAPTER 1 INTRODUCTION .............................. 1 2 PRELIMINARIES .............................. 7 2.1 Classi¯cation of p-groups with Derived Subgroup of Order p .... 7 2.2 Irreducible Characters of p-groups ................... 12 3 ISOMORPHIC CHARACTER TABLES .................. 18 4 THE NUMBER OF CHARACTER TABLES ............... 22 4.1 Character Tables of p-groups with Derived Subgroup of Order p .. 22 4.2 (m,n,p)-admissible Triples ....................... 39 4.3 Equivalent Kernels ........................... 41 4.4 Equivalent Images ........................... 42 4.5 Double Cosets .............................. 45 ¹ ¹ 4.6 Characterizations of Aut¼(B) and AutA(B) .............. 49 ¹ ¹ 4.7 The Number of (AutA(B); Aut¼(B)) Double Cosets ......... 55 4.8 Combinatorial Character Tables and Admissible Triples ....... 66 4.9 Combinatorial Character Tables and Blackburn Triples ....... 73 4.10 The Preimage of the Map cctn;m .................... 83 4.11 Classi¯cation of Character Tables ................... 93 5 THE LIMIT .................................. 99 5.1 Upper Bounds for the Size of the Preimage of the Map cctn;m ... 99 5.2 The Number of ¸'s ........................... 106 5.3 The Average Number of Distinct Parts of a Partition of an Integer . 112 5.4 lim NG(n; m)=NCT (n; m) ....................... 113 n!1 6 NUMERICAL RESULTS .......................... 126 REFERENCES ................................... 127 iv BIOGRAPHICAL SKETCH ............................ 128 v TABLE Table page 6{1 The number of groups and character tables ................. 126 vi Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Ful¯llment of the Requirements for the Degree of Doctor of Philosophy CHARACTER TABLES OF FINITE GROUPS By Adriana Nenciu August 2006 Chair: Alexandre Turull Major Department: Mathematics Precise formulas and estimates for the number of ¯nite p-groups up to isomorphism are known, but much less is known about the number of non-isomorphic character tables of such groups. Two character tables of ¯nite groups are isomor- phic if there exist a bijection for the irreducible characters and a bijection for the conjugacy classes that preserve all the character values. We give necessary and su±cient conditions for two ¯nite groups to have isomorphic character tables. In the case of ¯nite p-groups with derived subgroup of order p, we classify up to isomorphism their irreducible character tables, and estimate their number. The number of such character tables turns out to be considerably fewer than the corresponding number of groups. vii CHAPTER 1 INTRODUCTION The determination of all isomorphism types of groups of a ¯xed order has interested mathematicians since Cayley published his paper in 1854. In the last century a lot of e®ort was put into solving this problem and interesting results have been proved. In the particular case of p-groups, there are results that give an algorithm for constructing p-groups (see Eick and O'Brien [5] and Newman [10]), and results that give estimates for f(n; p), the number of groups of order pn (see Blackburn [3], Higman [7] and Sims [11]). G. Higman [7] proved that f(n; p) = pAn3 where A depends on n and p and 2 2 ¡ " · A · + " 27 n 15 n and lim "n = 0. Later S. R. Blackburn [3] gave upper and lower bounds for the n!1 number of p-groups in an isoclinism class: m Theorem 2 ([3]) Let © be an isoclinism class. De¯ne f©(p ) to be the number of groups of order pm in ©. Suppose that for all P 2 © we have jP=Z(P )P 0j = pa and jP 0 \ Z(P )j = pc. Then µ ¶(a+c)=2 m m f©(p ) (a+c)=2 k1 2 · m · k2m (log m) fab(p ) m where k1 and k2 are constants depending only on © and fab(p ) denotes the number of abelian groups of order pm. If progress has been made into classifying ¯nite groups, little is known about the classi¯cation up to isomorphism of character tables of ¯nite groups. 1 2 There are results about the uniqueness of character table (see Davydov [4] and Mattarei [9]). Mattarei's result gives a su±cient condition for two groups to have isomorphic character tables. Using quasi-Hopf algebras methods A. A. Davydov [4] gives necessary and su±cient conditions for two arbitrary ¯nite groups to have isomorphic character tables. Motivated by Davydov's result, in Chapter 3, we prove a similar result (see Proposition 3.0.14). Our statement and proof uses only methods from the representation theory of ¯nite groups. In a paper published in Journal of Algebra in 1999 [2], S. R. Blackburn gave a classi¯cation up to isomorphism of p-groups with derived subgroup of order p, for any prime p. The classi¯cation is independent of the prime p. Blackburn describes some combinatorial objects which can easily be enumerated and which are in one{to{one correspondence with the isomorphism classes of such groups. In this paper, we classify up to isomorphism the character tables of all p-groups whose derived subgroup has order p. We prove that there are two sets that are in bijection with the set of isomorphism classes of character tables of p-groups P with jP 0j = p. One set is algebraic and a priori it depends on the prime p. The second set is combinatorial and is independent of p. Thus, the number of character tables of all groups of order pn and derived subgroup of order p does not depend on the prime p, see Theorem 4.11.3 below. n Let NG(n) denote the number of non-isomorphic p-groups P with jP j = p 0 and jP j = p and NCT (n) denote the number of non-isomorphic character tables of these p-groups. By our and Blackburn's results these numbers do not depend on the prime p. We have that NG(n) is much larger than NCT (n). We prove (see Proposition 5.4.12 below) that NG(n) ¡ NCT (n) ¸ jP(n ¡ 3)j 3 where jP(n ¡ 3)j is the number of partitions of n ¡ 3. It then follows (see Theorem 5.4.13 below) that NG(n) ¡ NCT (n) lim p p = 1 n!1 e¼ 2=3 n n1+" for all " > 0. It is easily seen that, given a p-group P with jP j = pn and jP 0j = p, we 2m n¡1 have [P : Z(P )] = p for some positive integer 1 · m · b 2 c. We denote n 0 by NG(n; m) the number of non-isomorphic p-groups P with jP j = p , jP j = p 2m and [P : Z(P )] = p . We denote by NCT (n; m) the number of non-isomorphic character tables of these p-groups. With this notation, we obtain (see Theorem 5.4.11 below) NG(n; m) (2m)! lim = m : n!1 NCT (n; m) 2 m! n¡1 Let p be a prime and let m; n be positive integers such that 1 · m · b 2 c. n We denote by Pm;n;p the set of isomorphism classes of p-groups P with jP j = p , jP 0j = p and [P : Z(P )] = p2m. S. .R. Blackburn [2] proves that there exists a set Sn;m of integer vectors having certain properties (see De¯nition 4.2.6 below) and a bijection £m;n;p : Pm;n;p !Sn;m: The set Sn;m is independent of p; thus the number NG(n; m) = jPm;n;pj of these p-groups is independent of p. We denote by CT m;n;p the set of isomorphism classes of character tables of p-groups with derived subgroup of order p. We have a natural surjective map ctm;n;p : Pm;n;p ! CT m;n;p that associates to each isomorphism class of p-groups the isomorphism class of its character table. Using Proposition 3.0.14, we prove, in Corollary 4.1.11, that two groups P1;P2 2 Pm;n;p have isomorphic character tables if and only if there exist 4 0 0 group isomorphisms ® : P1=P1 ! P2=P2 and ¯ : Z(P1) ! Z(P2) such that the following diagram is commutative ¼¹1 / 0 Z(P1) P1=P1 ¯ ® ² ² ¼¹2 / 0 Z(P2) P2=P2 where¼ ¹1 and¼ ¹2 are the restrictions to Z(P1) and Z(P2) respectively if the 0 0 canonical projections P1 ³ P1=P1 and P2 ³ P2=P2 respectively. This result motivates the introduction, in Section 4.2, of (m; n; p)-admissible triples and their equivalence. We denote by Tm;n;p a full set of non-equivalent (m; n; p)-admissible triples. There is a natural map tm;n;p : Pm;n;p !Tm;n;p that associates to each isomorphism class of p-groups its corresponding (m; n; p)-admissible triple.