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 O2(L)
- The subgroup U = [O2(L), L] is an elementary abelian 2-group in the center of O2(L)
- L/O2(L) is quasisimple or of order 3
- L acts irreducibly on U/CU(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.
References
Aschbacher block Wikipedia(Text) CC BY-SA