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):

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.

References

• Aschbacher, Michael (1981), "Some results on pushing up in finite groups", Mathematische Zeitschrift, 177 (1): 61–80, doi:10.1007/BF01214339, ISSN 0025-5874, MR 0611470
• Aschbacher, Michael; Smith, Stephen D. (2004), The classification of quasithin groups. I Structure of Strongly Quasithin K-groups, Mathematical Surveys and Monographs, vol. 111, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3410-7, MR 2097623
• Foote, Richard (1980), "Aschbacher blocks", The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., vol. 37, Providence, R.I.: Amer. Math. Soc., pp. 37–42, MR 0604554
• Solomon, Ronald (1980), "Some results on standard blocks", The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., vol. 37, Providence, R.I.: Amer. Math. Soc., MR 0604555