Group in which every element is automorphic to its inverse

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

A group $G$ is termed a group in which every element is automorphic to its inverse if it satisfies the following equivalent conditions:

1. For every element $g \in G$, there is an automorphism $\sigma$ of $G$ such that $\sigma(g) = g^{-1}$.
2. There exists a group $K$ containing $G$ as a normal subgroup such that every element of $G$ is a real element of $K$: it is conjugate to its inverse in $K$.

Relation with other properties

Stronger properties

• Abelian group
• Ambivalent group
  • Rational group
  • Strongly ambivalent group
  • Rational-representation group
• Group whose automorphism group is transitive on non-identity elements
• Group in which every element is order-automorphic
• Group in which any two elements generating the same cyclic subgroup are automorphic

Facts

• Alternating group implies every element is automorphic to its inverse
• General linear group implies every element is automorphic to its inverse
• Projective general linear group implies every element is automorphic to its inverse
• Special linear group implies every element is automorphic to its inverse
• Projective special linear group implies every element is automorphic to its inverse

Metaproperties

Characteristic subgroups

This group property is characteristic subgroup-closed: any characteristic subgroup of a group with the property, also has the property
View characteristic subgroup-closed group properties]]

If every element of $G$ is automorphic to its inverse, then if $H$ is a characteristic subgroup of $G$, every element of $H$ is also automorphic to its inverse.