# Quasiautomorphism-invariant not implies 1-automorphism-invariant

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., quasiautomorphism-invariant subgroup) neednotsatisfy the second subgroup property (i.e., 1-automorphism-invariant subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about quasiautomorphism-invariant subgroup|Get more facts about 1-automorphism-invariant subgroup

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## Contents

## Statement

It is possible to have a group and a subgroup of such that is a quasiautomorphism-invariant subgroup of (i.e., is invariant under all Quasiautomorphism (?)s of ) but is *not* a 1-automorphism-invariant subgroup of (i.e., is *not* invariant under all 1-automorphisms of ).

## Related facts

## Facts used

## Proof

Let be an odd prime, let be the prime-cube order group:U(3,p), i.e., the unique non-abelian group of order and exponent , and let be the center of . is a cyclic subgroup of order in .

By fact (1), is quasiautomorphism-invariant in . However, there exist 1-automorphisms of that do not preserve . In fact, we can achieve any permutation of the cyclic subgroups of order using a 1-automorphism.