# Normal subgroup of ambivalent group implies every element is automorphic to its inverse

From Groupprops

## Statement

Suppose is an Ambivalent group (?) (i.e., every element of is conjugate to its inverse) and is a Normal subgroup (?) of . Then, is a Group in which every element is automorphic to its inverse (?): for every element , there is an automorphism of sending to its inverse.

## Related facts

- Normal subgroup of rational group implies any two elements generating the same cyclic subgroup are automorphic
- Every element is automorphic to its inverse is characteristic subgroup-closed
- Alternating group implies every element is automorphic to its inverse
- General linear group implies every element is automorphic to its inverse
- Special linear group implies every element is automorphic to its inverse
- Projective general linear group implies every element is automorphic to its inverse
- Projective special linear group implies every element is automorphic to its inverse