Normal not implies strongly potentially 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 subgroup) need not satisfy the second subgroup property (i.e., characteristic-potentially characteristic subgroup)
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Statement
Verbal statement
A normal subgroup need not be characteristic-potentially characteristic.
Statement with symbols
It is possible to have a group and a normal subgroup
of
such that there is no group
containing
in which both
and
are characteristic subgroups.
Facts used
- Characteristic-potentially characteristic implies normal-potentially characteristic
- Normal not implies normal-potentially characteristic
Proof
The proof follows directly from facts (1) and (2).