Minimal normal subgroup with order not dividing index is characteristic

Statement
A minimal normal subgroup of a finite group whose  order does not divide its  index is  characteristic.

Related facts

 * Equivalence of definitions of finite characteristically simple group

Facts used

 * 1) Automorphisms preserve the property of being normal and hence of being minimal normal.
 * 2) uses::Normality is strongly intersection-closed
 * 3) uses::Product formula

Proof
Given: A finite group $$G$$, a minimal normal subgroup $$H$$ of $$G$$. The order of $$H$$ does not divide the index $$[G:H]$$ of $$H$$ in $$G$$. An automorphism $$\sigma$$ of $$G$$.

To prove: $$\sigma(H) = H$$.

Proof: