Characteristic not implies sub-(isomorph-normal characteristic) in finite
From Groupprops
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., sub-(isomorph-normal characteristic) subgroup)
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Contents
Statement
We can have a finite group and a characteristic subgroup
of
such that
is not sub-(isomorph-normal characteristic) in
. In other words, there is no ascending chain of subgroups from
to
such that each member is an isomorph-normal characteristic subgroup of its successor.
Related facts
- Characteristic not implies sub-isomorph-free in finite
- Characteristic not implies isomorph-free in finite
Proof
The center of a non-abelian group of odd prime cube order
Further information: Prime-cube order group:U3p
Let be an odd prime. Let
be the non-abelian group of order
and exponent
. Let
be the center of
. Then, we have:
-
is characteristic in
.
-
is not sub-(isomorph-normal characteristic) in
: In fact, no proper subgroup of
containing
is isomorph-normal and characteristic.
itself is not isomorph-normal, because there are other cyclic groups of order
. Moreover,
is a maximal characteristic subgroup of
, so there is no other possibility.