Lifts of Partial Characters With Cyclic Defect groups

Abstract

We count the number of lifts of an irreducible π -partial character that lies in a block with a cyclic defect group.2010 Mathematics subject classification: primary 20C20; secondary 20C15.

Bounding the Diameter of the Brauer Graph of a Block of a Solvable Group

“…Write M = O p (G) and suppose that B covers the character α ∈ Irr(M ). If α is not invariant in G, then induction is a Brauer graph preserving bijection from a block B 0 of the stabilizer G α of α with defect group D to B (see, for example, [1]), and we are done by induction on |G| when α is not invariant in G.…”

Counting characters in blocks of solvable groups with abelian defect group

“…One can show that if (C G (D), b) is a root of B (see [3] for a discussion of roots of π-blocks with abelian defect group), then (D, b) is a B-subgroup, and thus [3] for more details of this argument, here we are essentially repeatedly applying Lemma 5.1 of [3] Notice that in the "classical" case (where π is the complement of the prime p), Theorem 4.1 of [10] shows that the conditions in Theorem 2 imply that B is nilpotent. In the π-case, we see that the "hard" direction of Theorem 1 (which required the use of a large orbit theorem) shows that the conditions of Theorem 2 imply that D is central in N V (D), though we do not know yet if this is equivalent to B being nilpotent.…”