Proposition 4

Let G be a group, H a characteristic (resp. normal) subgroup of G and K a characteristic subgroup of H. Then K is a characteristic (resp. normal) subgroup of G.

The restriction to H of an automorphism (resp. inner automorphism) of G is an automorphism of H and therefore leaves K invariant.

Let G be a group, A $\subset$ G and $b \in G$. b is said to normalize A if $bAb^{-1} = A$; b is said to centralize A if, for all $a \in A$, $bab^{-1} = a$. Let A and B be subsets of G; B is said to normalize (resp. centralize) A if every element of B normalizes (resp. centralizes) A.

The set of $g \in G$ which normalize (resp. centralize) A is called the normalizer (resp. centralizer) of A (cf. § 1, no. 5, Definition 9); it is often denoted by $N_G(A)$ or simply $N(A)$ (resp. $C_G(A)$ or $C(A)$). It is a subgroup of G. When A is a subgroup of G, $N_G(A)$ may be characterized as the largest subgroup of G which contains A and in which A is normal.