# 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

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

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

- For every element , there is an automorphism of such that .
- There exists a group containing as a normal subgroup such that every element of is a real element of : it is conjugate to its inverse in .

## Relation with other properties

### Stronger properties

- Abelian group
- Ambivalent 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 is automorphic to its inverse, then if is a characteristic subgroup of , every element of is also automorphic to its inverse.