Quasisimple group
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Contents
Definition
Symbol-free definition
A group is said to be quasisimple if it is perfect and its inner automorphism group is simple.
Definition with symbols
A group is said to be quasisimple if both the following hold:
- is perfect, that is,
- The inner automorphism group of is a simple group, that is, is simple (where denotes the center of ).
Classification
Further information: Classification of finite simple groups
The finite quasisimple groups can be completely classified in terms of the finite simple non-abelian groups. Specifically, for each finite simple non-abelian group , consider the Schur multiplier and the corresponding Schur covering group (the unique universal central extension) with its map to . The quasisimple groups with inner automorphism group are precisey the groups such that the map can be factored in terms of a surjective map and a surjective map .
In particular, by the fourth isomorphism theorem, these correspond precisely to the quotient groups (and hence to the subgroups, because of abelianness) of , which is the Schur multiplier of . In particular, for each finite simple non-abelian groups, there are finitely many quasisimple groups associated with it.
Examples
Finite simple non-abelian group | Order | Schur multiplier | Schur covering group | Quasisimple groups with this simple group as the inner automorphism group |
---|---|---|---|---|
alternating group:A5 | 60 | cyclic group:Z2 | special linear group:SL(2,5) | alternating group:A5, special linear group:SL(2,5) |
projective special linear group:PSL(3,2) (isomorphic to ) | 168 | cyclic group:Z2 | special linear group:SL(2,7) | projective special linear group:PSL(3,2), special linear group:SL(2,7) |
alternating group:A6 | 360 | cyclic group:Z6 | Schur cover of alternating group:A6 | alternating group:A6, special linear group:SL(2,9), triple cover of alternating group:A6, Schur cover of alternating group:A6 |
projective special linear group:PSL(2,8) | 504 | trivial group | projective special linear group:PSL(2,8) | projective special linear group:PSL(2,8) |
projective special linear group:PSL(2,11) | 660 | cyclic group:Z2 | special linear group:SL(2,11) | projective special linear group:PSL(2,11), special linear group:SL(2,11) |
Relation with other properties
Stronger properties
Weaker properties
- Inner-simple group: A group whose inner automorphism group is simple
- Perfect group
- Group in which every proper normal subgroup is central: For full proof, refer: Proper and normal in quasisimple implies central
- Group in which every normal subgroup is a central factor
- T-group
- Directly indecomposable group
- Splitting-simple group
- Group in which every endomorphism is trivial or an automorphism is weaker than the property of being a finite quasisimple group. For full proof, refer: Finite quasisimple implies every endomorphism is trivial or an automorphism
Facts
- The derived subgroup of an inner-simple group is quasisimple
- Any normal subgroup of a quasisimple group is either the whole group, or is contained inside the center For full proof, refer: Proper and normal in quasisimple implies central
References
Textbook references
- Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754^{More info}, Page 156 (definition in paragraph)