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. 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