# Group in which every element is automorphic to its inverse

## Contents

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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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## 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 $K$.

## Metaproperties

### Characteristic subgroups

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

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.