Normal not implies subgroup-conjugating

This article gives the statement and possibly, proof, of a non-implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., normal automorphism) need not satisfy the second automorphism property (i.e., subgroup-conjugating automorphism)
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Statement

A normal automorphism of a group (an automorphism that preserves all the normal subgroups) need not be subgroup-conjugating -- it need not send every subgroup to a conjugate subgroup.

Related facts

Stronger facts

• Class-preserving not implies subgroup-conjugating: An automorphism of a group that preserves conjugacy classes of elements need not send every subgroup to a conjugate subgroup.

Facts used

1. Class-preserving not implies subgroup-conjugating
2. Class-preserving implies normal

Proof

The proof follows directly by piecing together facts (1) and (2).