Blocks with extraspecial defect groups of finite quasisimple groups

Abstract

We classify all blocks of finite quasisimple groups with extraspecial defect groups.

Blocks of Finite Groups and Their Invariants

“…12 Let B be a p-block of a finite group where p is an odd prime, and let .u; b u / be a B-subsection such that l.b u / D 1 and b u has defect d . Moreover, let F be the fusion system of B, and let jAut F .hui/j D p s r where p − r and s 0.…”

“…Proposition 6. 12 Let F be a fusion system on a p-group P . If Q Ä P is Fessential of rank at most 3, then jN P .Q/ W Qj D p. Moreover, if p D 2, then…”

“…Moreover, many of the sporadic groups can be handled with GAP [266]. Information on blocks with specific properties can be found in articles by An and Eaton [7,[12][13][14].…”

“…Proposition 10. 12 Let P be a bicyclic 2-group such that P 0 Š C 2 2 r for some r 2. Then every fusion system on P is nilpotent.…”

“…Now we discuss the various possibilities for S , by making use of the classification of finite simple groups. In each case we apply [12].…”