# Normalizer of a subset

$M$ of a group $G$ in a subgroup $H$ of $G$

The set

$$N _ {H} ( M) = \ \{ {h } : {h \in H , h ^ {-} 1 M h = M } \} ,$$

that is, the set of all elements $h$ of $H$ such that $h ^ {-} 1 m h$( the conjugate of $m$ by $h$) for every $m \in M$ also belongs to $M$. For any $M$ and $H$ the normalizer $N _ {H} ( M)$ is a subgroup of $H$. An important special case is the normalizer of a subgroup of a group $G$ in $G$. A subgroup $A$ of a group $G$ is normal (or invariant, cf. Invariant subgroup) in $G$ if and only if $N _ {G} ( A) = G$. The normalizer of a set consisting of a single element is the same as its centralizer. For any $H$ and $M$ the cardinality of the class of subsets conjugate to $M$ by elements of $H$( that is, subsets of the form $h ^ {-} 1 M h$, $h \in H$) is equal to the index $| H : N _ {H} ( M) |$.

#### References

 [1] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)