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Normalizer of a subset

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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