Family automorphism - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In symbols, an automorphism $σ$ of a group $G$ is a family automorphism if, for all $x \in G$ , the subgroups generated by $x$ and $σ(x)$ 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. In Mathematics, in the realm of Group theory, a power automorphism of a group is an Automorphism that takes each Subgroup of the group Also, every class automorphism is a family automorphism. In Mathematics, in the realm of Group theory, a class automorphism is an Automorphism of a group that sends each element to within its Conjugacy
Every family automorphism restricts as an automorphism to normal subgroups. Hence, every family automorphism is a quotientable automorphism. In Mathematics, in the realm of Group theory, a quotientable automorphism of a group is an automorphism that takes every Normal subgroup to within
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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