Difference between revisions of "Normalizer of a subset"
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| + | '' $ M $ | ||
| + | of a [[Group|group]] $ G $ | ||
| + | in a subgroup $ H $ | ||
| + | of $ G $'' | ||
The set | The set | ||
| − | + | $$ | |
| + | N _ {H} ( M) = \ | ||
| + | \{ {h } : {h \in H , h ^ {-} 1 M h = M } \} | ||
| + | , | ||
| + | $$ | ||
| − | that is, the set of all elements | + | 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|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|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==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR></table> | ||
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| − | |||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.J.S. Robinson, "A course in the theory of groups" , Springer (1980)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.J.S. Robinson, "A course in the theory of groups" , Springer (1980)</TD></TR></table> | ||
Latest revision as of 08:03, 6 June 2020
$ 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
Normalizer of a subset. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normalizer_of_a_subset&oldid=48020
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