Normalizer-relatively normal subgroup
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This article describes a property that can be evaluated for a triple of a group, a subgroup of the group, and a subgroup of that subgroup.
View other such properties
Definition
Suppose are groups. We say that
is a normalizer-relatively normal subgroup of
with respect to
if the following equivalent conditions are satisfied:
- Whenever
is such that
is normal in
,
is also normal in
.
- The normalizer
is contained in the normalizer
.
-
is normal in
.
Relation with other properties
Stronger properties
- Middle-characteristic subgroup
- Strongly closed subgroup
- Weakly closed subgroup: For full proof, refer: Weakly closed implies normalizer-relatively normal