Stability group of subnormal series of finite group has no other prime factors
From Groupprops
This article states and (possibly) proves a fact about a finite group and a Coprime automorphism group (?): a subgroup of the automorphism group whose order is relatively prime to the order of the group itself.
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Contents
Statement
Let be a finite group, and consider a subnormal series of :
Then, any prime factor of the order of the stability group of this subnormal series, must divide the order of . In other words, no non-identity stability automorphism of this subnormal series can have order relatively prime to the order of .
Related facts
Generalizations
Particular cases
Converse
The converse to the statement is not true: we can have a subgroup of the automorphism group of , with no other prime factors to its order, which cannot be realized as the stability group of any subnormal series. However, the converse is true if we restrict ourselves to a group of prime power order:
- Centralizer-commutator product decomposition for finite nilpotent groups
- Centralizer-commutator product decomposition for finite groups
- Burnside's theorem on coprime automorphisms and Frattini subgroup
- Centralizer of coprime automorphism in homomorphic image equals image of centralizer
Facts used
References
Textbook references
- Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, ^{More info}, Page 18, Theorem 1.6.3 (Section 1.6)