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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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



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.