Normalizer of a subset
From Encyclopedia of Mathematics
of a group
in a subgroup
of
The set
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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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normalizer_of_a_subset&oldid=14809
Normalizer of a subset. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normalizer_of_a_subset&oldid=14809
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article