Quasiautomorphism-invariant not implies 1-automorphism-invariant
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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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 ).
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.