Automorph-conjugacy is transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
|

Statement

Property-theoretic statement

The subgroup property of being automorph-conjugate is transitive.

Symbolic statement

Let $H$ be an automorph-conjugate subgroup of $K$ , and $K$ be an automorph-conjugate subgroup of $G$ . Then, $H$ is an automorph-conjugate subgroup of $G$ .

Proof

Suppose $H$ is an automorph-conjugate subgroup of $K$ , and $K$ is an automorph-conjugate subgroup of $G$ . We want to show that $H$ is an automorph-conjugate subgroup of $G$ .

For this, pick any automorphism $σ$ of $H$ . Clearly, $\sigma(H) \le \sigma(K)$ , and since $K$ is automorph-conjugate subgroup of $G$ , there exists $g \in G$ such that $g σ(K) g - 1 = K$ . Thus, $c_g \circ \sigma$ (conjugation by $g$ , composed with $σ$ ), gives an automorphism of $K$ . Since $H$ is automorph-conjugate inside $K$ , there exists $h \in K$ such that $g σ(H) g - 1 = h H h - 1$ . Rearranging, we see that $σ(H) = g - 1 h H h - 1 g$ , a conjugate fo $H$ .

Automorph-conjugacy is transitive

From Groupprops

Contents

Statement

Property-theoretic statement

Symbolic statement

Proof

Views

Personal tools

Search

Navigation

lookup

Credits

request/feedback

Toolbox