Normal-potentially characteristic subgroup
From Groupprops
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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This term is related to: potentially characteristic subgroups characterization problem
View other terms related to potentially characteristic subgroups characterization problem | View facts related to potentially characteristic subgroups characterization problem
Definition
A subgroup of a group
is termed normal-potentially characteristic in
if there exists a group
containing
such that:
-
is a normal subgroup of
.
-
is a characteristic subgroup of
.
Formalisms
BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
In terms of the upper-hook operator
Given two subgroup properties and
, the upper-hook operator of
and
is defined as the following property
: a subgroup
of a group
has property
if there exists a group
containing
such that
has property
in
and
has property
in
.
The property of being semi-strongly potentially characteristic is thus obtained by applying the upper-hook operator to the properties characteristic subgroup and normal subgroup.