# Characteristic subset of a group

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.