Aschbacher block
# Aschbacher block

In mathematical finite group theory, a block, sometimes called Aschbacher block, is a subgroup giving an obstruction to Thompson factorization and pushing up. Blocks were introduced by Michael Aschbacher.

## Definition

A group L is called short if it has the following properties (Aschbacher & Smith 2004, definition C.1.7):

* L has no subgroup of index 2
* The generalized Fitting subgroup F*(L) is a 2-group $O_2(L)$
* The subgroup U = $[O_2(L),$ L] is an elementary abelian 2-group in the center of $O_2(L)$
* $L/O_2(L)$ is quasisimple or of order 3
* L acts irreducibly on $U/C_U(L)$

An example of a short group is the semidirect product of a quasisimple group with an irreducible module over the 2-element field $F_2$

A block of a group G is a short subnormal subgroup.

