Family automorphism

In mathematics, in the realm of group theory, an automorphism of a group is termed a family automorphism if it takes every element to an element generating a conjugate subgroup. In symbols, an automorphism <math>\sigma</math> of a group <math>G</math> is a family automorphism if, for all <math>x \in G</math>, the subgroups generated by <math>x</math> and <math>\sigma(x)</math> are conjugate.
Relations with other properties:
* Every subgroup-conjugating automorphism (that is, automorphism that sends each subgroup to a conjugate) is a family automorphism. This is because family automorphisms are precisely the automorphisms that sends cyclic subgroups to their conjugates. In particular, every power automorphism (an automorphism that restricts to an automorphism on each subgroup) is a family automorphism. Also, every class automorphism is a family automorphism.
* Every family automorphism restricts as an automorphism to normal subgroups. Hence, every family automorphism is a quotientable automorphism.
* An automorphism is a family automorphism if and only if it extends to an inner automorphism for evey representation over the rational numbers.

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