A Note on p-Blocks of a Finite Group (original) (raw)

Blocks of small defect in alternating groups and squares of Brauer character degrees

Journal of Group Theory, 2017

Let p be a prime. We show that other than a few exceptions, alternating groups will have p-blocks with small defect for p equal to 2 or 3. Using this result, we prove that a finite group G has a normal Sylow p-subgroup P and G / P {G/P} is nilpotent if and only if φ ⁢ ( 1 ) 2 {\varphi(1)^{2}} divides | G : ker ( φ ) | {|G:{\rm ker}(\varphi)|} for every irreducible Brauer character φ of G.

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.

The influence of p-regular class sizes on normal subgroups

Journal of Group Theory, 2013

Let G be a finite group and N be a normal subgroup of G and suppose that the p-regular elements of N have exactly two G-conjugacy class sizes. It is shown that N is solvable and that if H is a p-complement of N , then either H is abelian or H is the product of an r-group for some prime r 6 D p and a central subgroup of G.

Normal subgroups and p-regular G-class sizes

Journal of Algebra, 2011

Let G be a finite p-solvable group and N be a normal subgroup of G. Suppose that the p-regular elements of N have exactly two G-conjugacy class sizes. In this paper it is shown that, if H is a p-complement of N, then either H is abelian or H is a product of a q-group for some prime q = p and a central subgroup of G.

Blocks and subgroups

Journal of Algebra, 1972

Let G be a finite group and p a prime. The principal p-block of G is denoted by Bo(G) and as usual OP'(G) is the subgroup of G generated by all p-elements. We prove THEOREM A. If!f; is an irreducible ordinary or Brauer character in Bo(G) and He G with p 11 G : H I, then the restriction !f;H has a constituent in Bo(H). Furthermore, ifOP'(G) C H then every irreducible constituent of!f;H lies in Bo(H). If X is an ordinary character of G and b is a block of H C G we denote by Xb the sum of those constituents of XH which lie in b. In this situation we prove THEOREM B. Let X be an ordinary irreducible character of G and let b be a block of H C G. Then 1 G : H 1 Xb(I)/X(I) = ex is a p-local integer. Furthermore, if X E Bo(G), then ex =0 0 mod p unless p 11 G : HI and b = Bo(H) in which case ex =0 1 G : H 1 mod p. Although the conclusions of these two theorems overlap, neither seems to imply the other. Both theorems will be proved by studying indecomposable modules, and the Green Correspondence will be used heavily for Theorem A. 1. Let K be a complete p-adic field (characteristic 0) and let R be the ring of p-Iocal integers in K. Let (7T) be the principal maximal ideal of Rand * Work partially supported by NSF-GP-18640. t Work partially supported by NSF-GP-9572.

Counting characters in blocks of solvable groups with abelian defect group

arXiv (Cornell University), 2011

If G is a solvable group and p is a prime, then the Fong-Swan theorem shows that given any irreducible Brauer character ϕ of G, there exists a character χ ∈ Irr(G) such that χ o = ϕ, where o denotes the restriction of χ to the p-regular elements of G. We say that χ is a lift of ϕ in this case. It is known that if ϕ is in a block with abelian defect group D, then the number of lifts of ϕ is bounded above by |D|. In this paper we give a necessary and sufficient condition for this bound to be achieved, in terms of local information in a subgroup V determined by the block B. We also apply these methods to examine the situation when equality occurs in the k(B) conjecture for blocks of solvable groups with abelian defect group.

Modular Representation Theory of Finite Groups with T. I. Sylow p-Subgroups

Transactions of the American Mathematical Society, 1990

Let p be a fixed prime, and let G be a finite group with a T.I. Sylow p-subgroup P. Let N = NG(P) and let k(G) be the number of conjugacy classes of G. If z(G) denotes the number of p-blocks of defect zero, then we show in this article that z(G) = k(G)-Zc(A'). This result confirms a conjecture of J. L. Alperin. Its proof depends on the classification of the finite simple groups. Brauer's height zero conjecture and the Alperin-McKay conjecture are also verified for finite groups with a T.I. Sylow p-subgroup.