Stability group of subnormal series of finite group has no other prime factors

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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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Statement

Let G be a finite group, and consider a subnormal series of G:

\{ e \} \triangleleft H_1 \triangleleft H_2 \triangleleft \dots \triangleleft H_{n-1} \triangleleft H_n = G

Then, any prime factor of the order of the stability group of this subnormal series, must divide the order of G. In other words, no non-identity stability automorphism of this subnormal series can have order relatively prime to the order of G.

Related facts

Generalizations

Particular cases

Converse

The converse to the statement is not true: we can have a subgroup of the automorphism group of G, 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 G a group of prime power order:

Other related facts

Facts used

  1. Centralizer of coprime automorphism in homomorphic image equals image of centralizer

References

Textbook references