# Derived subgroup not is quasiautomorphism-invariant

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., commutator subgroup) doesnotalways satisfy a particular subgroup property (i.e., {{{property}}}Property "Proves property satisfaction of" (as page type) with input value "{{{property}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.)

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Statement

It is possible to have a group such that the commutator subgroup (or derived subgroup) is not a quasiautomorphism-invariant subgroup, i.e., there exists a Quasiautomorphism (?) of such that .

## Related facts

- Characteristic not implies quasiautomorphism-invariant
- Quasiautomorphism-invariant not implies 1-automorphism-invariant
- Center is quasiautomorphism-invariant
- Center not is 1-automorphism-invariant

## Proof

`Further information: inner automorphism group of wreath product of groups of order p`

Suppose is a prime number greater than . Let be the group isomorphic to the inner automorphism group of the wreath product of two groups of order . is a group of order with an elementary abelian normal subgroup of order , an element of order acting on it from outside, and every non-identity element of has order .

Consider the commutator subgroup . is a group of order , contained inside the elementary abelian normal subgroup.

we can construct a quasiautomorphism of that does not preserve as follows: the restriction of to is an automorphism of that fixes the center (which is cyclic of order ) but does not send to itself, and fixes every element of outside . Note that we need to ensure that is strictly bigger than , which is necessary to be able to construct a with the desired specifications.