Derived subgroup not is quasiautomorphism-invariant

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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) does not always satisfy a particular subgroup property (i.e., {{{property}}}
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Statement

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

Related facts

Proof

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

Suppose p is a prime number greater than 3. Let G be the group isomorphic to the inner automorphism group of the wreath product of two groups of order p. G is a group of order p^p with an elementary abelian normal subgroup N of order p^{p-1}, an element of order p acting on it from outside, and every non-identity element of G has order p.

Consider the commutator subgroup H = [G,G]. H is a group of order p^{p-2}, contained inside the elementary abelian normal subgroup.

we can construct a quasiautomorphism \sigma of G that does not preserve H as follows: the restriction of \sigma to N is an automorphism of N that fixes the center Z(G) (which is cyclic of order p) but does not send H to itself, and \sigma fixes every element of G outside H. Note that we need p > 3 to ensure that H is strictly bigger than Z(G), which is necessary to be able to construct a \sigma with the desired specifications.