Characteristic Subgroup of Normal Subgroup is Normal

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $G$ be a group.

Let $N\leq G$ be normal.

Let $H\leq N$ be characteristic.


Then $H$ is normal in $G$.


Proof

Let $g \in G$.

Because $N$ is normal, conjugation by $g$ is an automorphism of $N$.

Because $H$ is characteristic in $N$, $g H g^{-1} = H$.

Thus $H$ is normal in $G$.

$\blacksquare$


Sources