Complete not implies ambivalent

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

A complete group (i.e., a centerless group in which every automorphism is an inner automorphism) need not be an ambivalent group (i.e., a group in which every element is conjugate to its inverse).

Facts used

  1. Holomorph of cyclic group of odd prime order is complete
  2. Ambivalence is quotient-closed
  3. Characteristically simple and non-abelian implies automorphism group is complete
  4. Automorphism group of simple non-abelian group need not be ambivalent

Proof

Proof using facts (1) and (2)

Further information: Holomorph of Z5

By fact (1), the holomorph of any cyclic group of odd prime order is complete. Let p = 5 and consider the holomorph of the cyclic group of order p. This is a group of order 5 \cdot 4 = 20, with the automorphism group a cyclic group of order 4.

On the other hand, if this holomorph were an ambivalent group, then, by fact (2), any quotient of it would also be ambivalent. But the quotient by the cyclic normal subgroup of order five is its automorphism group, which is a cyclic group of order four, which is not an ambivalent group because the generator and its inverse are not conjugate.

Proof using facts (3) and (4)

The proof follows directly from facts (3) and (4).