Quasiautomorphism-invariant not implies 1-automorphism-invariant

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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) need not satisfy the second subgroup property (i.e., 1-automorphism-invariant subgroup)
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Statement

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

Related facts

Facts used

  1. Center is quasiautomorphism-invariant

Proof

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

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