Modular Representation Theory of Finite Groups with T. I. Sylow p-Subgroups (original) (raw)

Defect Zero p-Blocks for Finite Simple Groups

1996

We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a p-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p−blocks remained unclassified were the alternating groups An. Here we show that these all have a p-block with defect 0 for every prime p ≥ 5. This follows from proving the same result for every symmetric group Sn, which in turn follows as a consequence of the t-core partition conjecture, that every non-negative integer possesses at least one t-core partition, for any t ≥ 4. For t ≥ 17, we reduce this problem to Lagrange's Theorem that every non-negative integer can be written as the sum of four squares. The only case with t < 17, that was not covered in previous work, was the case t = 13. This we prove with a very different argument, by interpreting the generating function for t-core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne's Theorem (née the Weil Conjectures). We also consider congruences for the number of p-blocks of Sn, proving a conjecture of Garvan, that establishes certain multiplicative congruences when 5 ≤ p ≤ 23. By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime p and positive integer m, the number of p−blocks with defect 0 in Sn is a multiple of m for almost all n. We also establish that any given prime p divides the number of p−modularly irreducible representations of Sn, for almost all n.

On the characters of ppp-solvable groups

Transactions of the American Mathematical Society, 1961

Introduction. In the theory of group characters, modular representation theory has explained some of the regularities in the behavior of the irreducible characters of a finite group; not unexpectedly, the theory in turn poses new problems of its own. These problems, which are asked by Brauer in [4], seem to lie deep. In this paper we look at the situation for solvable groups. We can answer some of the questions in [4] for these groups, and in doing so, obtain new properties for their characters. Finite solvable groups have recently been the object of much investigation by group theorists, especially with the end of relating the structure of such groups to their Sylow /»-subgroups. Our work does not lie quite in this direction, although we have one result tying up arithmetic properties of the characters to the structure of certain /»-subgroups. Since the prime number p is always fixed, we can actually work in the more general class of /»-solvable groups, and shall do so. Let © be a finite group of order g = pago, where p is a fixed prime number, a is an integer ^0, and ip, go) = 1. In the modular theory, the main results of which are in [2; 3; 5; 9], the characters of the irreducible complex-valued representations of ©, or as we shall say, the irreducible characters of ®, are partitioned into disjoint sets, these sets being the so-called blocks of © for the prime p. Each block B has attached to it a /»-subgroup 35 of © determined up to conjugates in @, the defect group of the block B. If 35 has order pd, in which case we say B has defect d, and if Xn is an irreducible character in B, then the degree of Xk-¡s divisible by p to the exponent a-d+e", where the nonnegative integer e" is defined as the height of Xm-Now let © be a /»-solvable group, that is, © has a composition series such that each factor is either a /»-group or a /»'-group, a /»'-group being one of order prime to p. The following is then true: Let B be a block of © with defect group 3). If 35 is abelian, then every character Xu in B has height 0. Conversely, if B is the block containing the 1-character, and if every character in B has height 0, then 35 is abelian. In particular, this gives a necessary and sufficient condition for the Sylow /»-subgroups of © to be abelian. In the general modular theory of finite groups, each irreducible character X" can be decomposed into a sum of irreducible modular characters <£p of ®,

On the character degrees of Sylow p-subgroups of Chevalley groups of type E

Forum Mathematicum, 2000

Let Fq be a field of characteristic p with q elements. It is known that the degrees of the irreducible characters of the Sylow p-subgroup of GLn(Fq) are powers of q, see Isaacs . On the other hand Sangroniz showed that this is true for a Sylow p-subgroup of a classical group defined over Fq if and only if p is odd. For the classical groups of Lie type B, C and D the only bad prime is 2. For the exceptional groups there are others. In this paper we construct irreducible characters for the Sylow p-subgroups of the Chevalley groups D 4 (q) with q = 2 f of degree q 3 /2. Then we use an analogous construction for E 6 (q) with q = 3 f to obtain characters of degree q 7 /3, and for E 8 (q) with q = 5 f to obtain characters of degree q 16 /5. This helps to explain why the primes 2, 3 and 5 are bad for the Chevalley groups of type E in terms of the representation theory of the Sylow p-subgroup.

Cosets of Sylow p -subgroups and a question of Richard Taylor

Journal of Algebra, 2014

We prove that for any odd prime p, there exist infinitely many finite simple groups S containing a Sylow p-subgroup P of S such that some coset gP of P in S consists of elements whose order is divisible by p. This allows us to answer a question of Richard Taylor related to whether certain Galois representations are automorphic.

Certain relations between p-regular class sizes and the p-structure of p-solvable groups

Journal of the Australian Mathematical Society, 2004

Let G be a finite p-solvable group for a fixed prime p. We study how certain arithmetical conditions on the set of p-regular conjugacy class sizes of G influence the p-structure of G. In particular, the structure of the p-complements of G is described when this set is {1, m, n} for arbitrary coprime integers m, n > 1. The structure of G is determined when the noncentral p-regular class lengths are consecutive numbers and when all of them are prime powers.

Simple groups of order p · 3a · 2b

Journal of Algebra, 1970

In this paper some local group theoretic properties of a simple group G of order p * 3@ * 2b are found. These are applied in a later paper to show there are no simple groups of order 7 * 3" * 2b other than the three well-known ones. R. Brauer [4] has shown there are only the three known simple groups LI, , A, , and O,(3) of order 5 .3" * 2b. His treatment uses modular character theory especially for the prime 5. It follows from J. Thompson's X-group paper [9] that if G is a simple group of order p + 3" * 2*, then p = 5,7, 13, or 17. In our later treatment of the case p '=-T 7 we seem to need the results of the N-group paper itself. This present paper prepares the way.

A solvability criterion for finite groups related to the number of Sylow subgroups

Communications in Algebra, 2020

Let G be a finite group and let pðGÞ be the set of primes dividing the order of G. For each p 2 pðGÞ, the Sylow theorems state that the number of Sylow p-subgroups of G is equal to kp þ 1 for some non-negative integer k. In this article, we characterize non-solvable groups G containing at most p 2 þ 1 Sylow p-subgroups for each p 2 pðGÞ: In particular, we show that each finite group G containing at most ðp À 1Þp þ 1 Sylow p-subgroups for each p 2 pðGÞ is solvable.