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

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 $$G$$.

Generalizations

 * Left-stability group of subgroup series of finite group has no other prime factors

Particular cases

 * Stability group of subnormal series of p-group is p-group

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:


 * p-group of automorphisms of p-group is contained in stability group of some normal series

Other related facts

 * 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

 * 1) uses::Centralizer of coprime automorphism in homomorphic image equals image of centralizer

Textbook references

 * , Page 18, Theorem 1.6.3 (Section 1.6)