Let \(G\) be a group, and let \(H \le K \le G\) be subgroups. Prove that if \(H\) is a characteristic subgroup of \(K\) (\(H \operatorname{char} K\)) and \(K\) is a characteristic subgroup of \(G\) (\(K \operatorname{char} G\)), then \(H\) is a characteristic subgroup of \(G\) (\(H \operatorname{char} G\)).
Solution
Let \(\varphi\in\operatorname{Aut}(G)\). Since \(K\operatorname{char}G\), \[ \varphi(K)=K. \] Therefore the restriction \[ \varphi|_K:K\to K \] is an automorphism of \(K\). Since \(H\operatorname{char}K\), this restriction preserves \(H\): \[ \varphi(H)=(\varphi|_K)(H)=H. \] Because this holds for every automorphism \(\varphi\) of \(G\), we have \[ H\operatorname{char}G. \] Thus characteristicity is transitive.