Weight for a finite group

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Definition with symbols

let p be a prime and G a finite group. A weight of G is a pair (R,\phi) such that:

  • R is a p-subgroup of G, and in fact, R = O_p(N_G(R)) (viz it is the p-Sylow core of its normalizer N_G(R)).
  • \phi is an irreducible character of N_G(R), the restriction of \phi to R is trivial, and \phi belongs to a p-block of N_G(R)/R of defect zero.

Equivalence notion

Two weights are said to be equal if their is an inner automorphism of G taking one to the other.

Weights for a block

For B a p-block of G, the weight is said to eb a B-weight if B=b^G where b is the associated block on N_G(R)/R.