Aschbacher block explained

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 :

1. L has no subgroup of index 2
2. The generalized Fitting subgroup F*(L) is a 2-group O₂(L)
3. The subgroup U = [''O''₂(''L''), ''L''] is an elementary abelian 2-group in the center of O₂(L)
4. L/O₂(L) is quasisimple or of order 3
5. 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 F₂

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