Definition 4

Let G be a group and $x \in G$. The automorphism $y \mapsto xyx^{-1}$ is called the inner automorphism of G defined by x and is denoted by $\operatorname{Int} x$.

For $x, y \in G$, we also write $x^y = y^{-1}xy = (\operatorname{Int} y^{-1})(x)$.

A subgroup of G is normal if and only if it is stable under all inner automorphisms of G (§ 4, no. 4, Definition 5). A subgroup of G is called characteristic if it stable under all automorphisms of G. The centre of a group G is a characteristic subgroup (formula (2)).

The centre of a group G is not necessarily stable under all endomorphisms of G (Exercise 22). In particular, the centre of a group with operators is not necessarily a stable subgroup.