# 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) |

#### Comments

#### References

[a1] | D.J.S. Robinson, "A course in the theory of groups" , Springer (1980) |

**How to Cite This Entry:**

Normalizer of a subset.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Normalizer_of_a_subset&oldid=48020