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 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.