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
Michael George Aschbacher (born April 8, 1944) 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 turne ...
.
Definition
A group ''L'' is called short if it has the following properties :
#''L'' has no subgroup of index 2
#The
generalized Fitting subgroup In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup ''F'' of a finite group ''G'', named after Hans Fitting, is the unique largest normal nilpotent subgroup of ''G''. Intuitively, it represents the smallest ...
''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.
References
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*{{Citation , last1=Solomon , first1=Ronald , title=The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) , publisher=Amer. Math. Soc. , location=Providence, R.I. , series=Proc. Sympos. Pure Math. , mr=604555 , year=1980 , volume=37 , chapter=Some results on standard blocks
Finite groups