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Let be a group, and let
. The normalizer of in , written , is the set
A subgroup of is said to be self-normalizing if
.
is always a subgroup of , as it is the stabilizer of under the action
of on the set of all subsets of (or on the set of all subgroups of , if is a subgroup).
If is a subgroup of , then
.
If is a subgroup of , then is a normal subgroup of ; in fact, is the largest subgroup of of which is a normal subgroup. In particular, if
is a subgroup of , then is normal in if and only if .
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