# Normalizer of a subset

From Encyclopedia of Mathematics

* of a group in a subgroup of *

The set

that is, the set of all elements of such that (the conjugate of by ) for every also belongs to . For any and the normalizer is a subgroup of . An important special case is the normalizer of a subgroup of a group in . A subgroup of a group is normal (or invariant, cf. Invariant subgroup) in if and only if . The normalizer of a set consisting of a single element is the same as its centralizer. For any and the cardinality of the class of subsets conjugate to by elements of (that is, subsets of the form , ) is equal to the index .

#### 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. N.N. Vil'yams (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Normalizer_of_a_subset&oldid=14809

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098