Blocks of Defect Zero and Products of Elements of Orderp

In this paper we investigate general properties of Cartan invariants of a finite group G in characteristic 2. One of our results shows that the Cartan matrix of G in characteristic 2 contains an odd diagonal entry if and only if G contains a real element of 2-defect zero. We also apply these results to 2-blocks of symmetric groups and to blocks with normal or abelian defect groups. The second part of the paper deals with annihilators of certain ideals in centers of group algebras and blocks.

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“…(iii) The case where p = 2 and m = n = 1 is handled in [19,Proposition 4.1]. So we suppose that p is odd or that m > 1 or that n > 1.…”

Section: Group Algebrasmentioning

confidence: 99%

Cartan invariants and central ideals of group algebras

et al. 2006

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“…The precise definition will be given below. Motivated by the special case of group algebras [8,9], we show that Z 0 A ⊆ T 1 A ⊥ 2 ⊆ HA, so that (T 1 A ⊥ ) 2 fits nicely into the chain of ideals above. When p is odd then…”

Section: Introductionmentioning

confidence: 95%

Central ideals and Cartan invariants of symmetric algebras

et al. 2005

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In this paper, we investigate certain ideals in the center of a symmetric algebra A over an algebraically closed field of characteristic p > 0. These ideals include the Higman ideal and the Reynolds ideal. They are closely related to the p-power map on A. We generalize some results concerning these ideals from group algebras to symmetric algebras, and we obtain some new results as well. In case p = 2, these ideals detect odd diagonal entries in the Cartan matrix of A. In a sequel to this paper, we will apply our results to group algebras.

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“…Before our first result, we recall some results of [Murray 1999;Robinson 1983]. Let P be a Sylow p-subgroup of G. In [Robinson 1983], it is proved that the number of p-blocks of defect zero is the rank of a matrix S with entries in GF( p) defined as follows: The rows and columns of S are indexed by the conjugacy classes of p-regular elements y of G such that C G (y) is a p -group.…”

Section: Introductionmentioning

confidence: 99%

Involutions, weights andp-local structure

Robinson

2011

ANT

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We prove that for an odd prime p, a finite group G with no element of order 2 p has a p-block of defect zero if it has a non-Abelian Sylow p-subgroup or more than one conjugacy class of involutions. For p = 2, we prove similar results using elements of order 3 in place of involutions. We also illustrate (for an arbitrary prime p) that certain pairs (Q, y), with a p-regular element y and Q a maximal y-invariant p-subgroup, give rise to p-blocks of defect zero of N G (Q)/Q, and we give lower bounds for the number of such blocks which arise. This relates to the weight conjecture of J. L. Alperin. IntroductionInvolutions have played a crucial role in finite group theory for many decades. They also figure prominently in representation theory, both ordinary and modular. Examples of the former include their occurrence in finite reflection groups, and an example of the latter is that in characteristic 2, J. Murray proved in [2006] that the projective summands of the (characteristic 2) permutation module (under conjugation action) on the solutions of x 2 = 1 in G are (in bijection with) the real 2-blocks of defect zero.Involutions also influence representation theory in odd characteristic. It was proved by Brauer and Fowler in [1955] that when p is an odd prime, G has a pblock of defect zero if there is an involution t ∈ G that neither inverts nor centralizes any nontrivial p-element of G. This result was extended by T. Wada [1977], who proved that if there are r mutually nonconjugate involutions of G that neither invert nor centralize any nontrivial p-element of G, then G has at least r distinct p-blocks of defect zero. We prove here that when p = 2, elements of order 3 can play a role analogous to that played when p is odd by involutions in the results above: We prove that the number of 2-blocks of defect zero of G is at least as great as the number of conjugacy classes of elements of order 3 that normalize no nontrivial 2-subgroup of G.We also point out here that results of this nature can be combined with local group-theoretic analysis to prove that if p is an odd prime and G is a group without MSC2010: 20C20.

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Blocks of Defect Zero and Products of Elements of Orderp

Cited by 10 publications

References 14 publications

Cartan invariants and central ideals of group algebras

Cartan invariants and central ideals of group algebras

Central ideals and Cartan invariants of symmetric algebras

Involutions, weights andp-local structure

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