In mathematical finite group theory, a block, sometimes called Aschbacher block, is a subgroup giving an obstruction to Thompson factorization

Thompson factorization

In mathematical finite group theory, a Thompson factorization, introduced by , is an expression of some finite groups as a product of two subgroups, usually normalizers or centralizers of p-subgroups for some prime p....

 and pushing up. Blocks were introduced by Michael Aschbacher

Michael Aschbacher

Michael George Aschbacher is an American mathematician best known for his work on finite groups. He was a leading figure in the completion of the classification of finite simple groups in the 1970s and 1980s. It later turned out that the classification was incomplete, because the case of quasithin...

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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 = [O₂(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.