Characteristic not implies fully invariant in odd-order class two p-group
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 odd-order class two p-group. That is, it states that in a odd-order class two p-group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., fully invariant subgroup)
View all subgroup property non-implications | View all subgroup property implications
Statement
For any (odd) prime , there exists a
-group
of class two and a characteristic subgroup of this group that is not fully invariant.
The construction also works for , but for
, there are already examples of abelian groups with characteristic subgroups that are not fully invariant.
Related facts
- Characteristic not implies fully invariant in finite abelian group
- Characteristic equals fully invariant in odd-order abelian group
- Characteristic not implies fully invariant in class three maximal class p-group
- Center not is fully invariant in class two p-group
- Socle not is fully invariant in class two p-group
Proof
Let be an odd prime. Let
be any non-abelian group of order
with center
. There are two possibilities for
: a group of prime-square exponent, and a group of prime exponent. In both such groups, there is an element
of order
outside
.
Define where
is the cyclic group of order
with generator
. The center of
is the subgroup
.
Then:
-
is characteristic in
, because center is characteristic.
-
is not fully invariant in
: Consider the retraction with kernel
and with image generated by the element
. This is an endomorphism of
, but it does not send
to itself, since the element
gets sent to
, which is outside
.