Normal-potentially characteristic not implies characteristic
From Groupprops
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal-potentially characteristic subgroup) need not satisfy the second subgroup property (i.e., characteristic subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
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Statement
It is possible to have a group and a subgroup
such that both the following hold:
-
is not a characteristic subgroup of
.
- There exists a group
containing
as a normal subgroup and
as a characteristic subgroup.
Facts used
Proof
By fact (1), any subgroup of a finite abelian group can be realized as a characteristic subgroup in some bigger abelian group. Thus, if is a finite abelian group and
is a non-characteristic subgroup of
, there exists some abelian group
containing
in which
is characteristic. Note that
is normal in
, because
is abelian.
For instance, if we take to be the Klein four-group, and
as one of the subgroups of order two, then we can take
as a direct product of Z4 and Z2, containing
as one of its cyclic subgroups.