Characteristic subset of a group

This article defines a property of subsets of groups
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Definition

A subset $A$ of a group $G$ is termed a characteristic subset if it satisfies the following equivalent conditions:

1. For any automorphism $\sigma$ of $G$, $\sigma (A)\subseteq A$, i.e., $\sigma (a)\in A$ for all $a\in A$.
2. For any automorphism $\sigma$ of $G$, $\sigma (A)=A$.
3. For any automorphism $\sigma$ of $G$, the restriction of $\sigma$ to $A$ is a bijection from $A$ to itself.

Facts

• Characteristic subset generates characteristic subgroup