Subgroup-conjugating not implies class-preserving

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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., subgroup-conjugating automorphism) need not satisfy the second automorphism property (i.e., class-preserving automorphism)
View a complete list of automorphism property non-implications | View a complete list of automorphism property implications
Get more facts about subgroup-conjugating automorphism|Get more facts about class-preserving automorphism

Statement

A subgroup-conjugating automorphism of a group (i.e., an automorphism that sends every subgroup to a conjugate subgroup) need not be class-preserving: it need not send every element to a conjugate element.

Related facts

Stronger facts

Weaker facts

Converse

Facts used

  1. Universal power not implies class-preserving
  2. Universal power implies subgroup-conjugating

Proof

The proof follows directly from combining facts (1) and (2).