Normal core of normalizer
From Groupprops
This article defines a subgroup operator related to the subgroup property normal subgroup. By subgroup operator is meant an operator that takes as input a subgroup of a group and outputs a subgroup of the same group.
Definition
The normal core of normalizer of a subgroup of a group
is defined in the following equivalent ways:
- It is the largest normal subgroup of
that normalizes
. In other words, it is the largest normal subgroup of
contained in the normalizer
.
- It is the normal core of the normalizer
in
.
is contained in the normal core of normalizer of
if and only if
is a 2-subnormal subgroup of
. In this case, if
is the normal core of normalizer of
, then the ascending chain
is the unique fastest ascending subnormal series for
in
. Further information: 2-subnormal subgroup has a unique fastest ascending subnormal series