# Equivalent Definitions of Characteristic Subgroups. Center is Characteristic.

## Problem 246

Let $H$ be a subgroup of a group $G$. We call $H$ **characteristic** in $G$ if for any automorphism $\sigma\in \Aut(G)$ of $G$, we have $\sigma(H)=H$.

**(a)** Prove that if $\sigma(H) \subset H$ for all $\sigma \in \Aut(G)$, then $H$ is characteristic in $G$.

**(b)** Prove that the center $Z(G)$ of $G$ is characteristic in $G$.

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## Definition

Recall that an automorphism $\sigma$ of a group $G$ is a group isomorphism from $G$ to itself.

The set of all automorphism of $G$ is denoted by $\Aut(G)$.

## Proof.

### (a) If $\sigma(H) \subset H$ for all $\sigma \in \Aut(G)$, then $H$ is characteristic in $G$

Since $\sigma$ is an automorphism, the inverse $\sigma^{-1}$ is also an automorphism of $G$.

Hence, we have

\[\sigma^{-1}(H)\subset H\]
by the assumption.

Applying $\sigma$, we have

\[\sigma\sigma^{-1}(H) \subset \sigma(H).\]
Then we obtain

\begin{align*}

H&=\sigma \sigma^{-1}(H)\subset \sigma(H)\subset H.

\end{align*}

Since the both ends are $H$, the inclusion is in fact the equality.

Thus, we obtain

\[\sigma(H)=H,\]
and the subgroup $H$ is characteristic in the group $G$.

### (b) The center $Z(G)$ of $G$ is characteristic in $G$

By part (a), it suffices to prove that $\sigma(Z(G)) \subset Z(G)$ for every automorphism $\sigma \in \Aut(G)$ of $G$.

Let $x\in \sigma(Z(G))$. Then there exists $y \in Z(G)$ such that $x=\sigma(y)$.

To show that $x \in Z(G)$, consider an arbitrary $g \in G$.

Then since $\sigma$ is an automorphism, we have $G=\sigma(G)$.

Thus there exists $g’$ such that $g=\sigma(g’)$.

We have

\begin{align*}

xg &=\sigma(y)\sigma(g’)\\

&=\sigma(yg’) && \text{ (since $\sigma$ is a homomorphism)}\\

&=\sigma(g’y) && \text{ (since $y \in Z(G)$)}\\

&=\sigma(g’)\sigma(y) && \text{ (since $\sigma$ is a homomorphism)}\\

&=gx.

\end{align*}

Since this is true for all $g \in G$, it follows that $x \in Z(G)$, and thus

\[\sigma(Z(G)) \subset Z(G).\]
This completes the proof.

## Comment.

In some textbook, a subgroup $H$ of $G$ is said to be characteristic in $G$ if $\sigma(H) \subset H$ for all $\sigma \in \Aut(G)$.

Problem (a) implies that our definition of characteristic and this alternative definition are in fact equivalent.

## Related Question.

Read the post Basic properties of characteristic groups for more problems about characteristic subgroups.

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