Normalizer
A normalizer is a part of a group.
Let be a subset of a group
. An element
of
is said to normalize
if
. A subset
of
is said to normalize
if all its elements normalize
. The set of all elements of
that normalize
is called the normalizer of
. It is often denoted as
, or
, when there is no risk of confusion. It is evidently a subgroup of
; for
; if
normalize
, then
and if
, then
. Evidently,
.
When is a subgroup of
,
is the largest subgroup of
of which
is a normal subgroup.
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