Normalizer of a subset
of a group in a subgroup of
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 .
|||M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)|
|[a1]||D.J.S. Robinson, "A course in the theory of groups" , Springer (1980)|
Normalizer of a subset. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normalizer_of_a_subset&oldid=14809