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.

Basic Properties of Characteristic Groups
Definition (automorphism).
An isomorphism from a group $G$ to itself is called an automorphism of $G$.
The set of all automorphism is denoted by $\Aut(G)$.
Definition (characteristic subgroup).
A subgroup $H$ of a group $G$ is called characteristic in $G$ if for any $\phi […]

Group Homomorphism, Conjugate, Center, and Abelian group
Let $G$ be a group. We fix an element $x$ of $G$ and define a map
\[ \Psi_x: G\to G\]
by mapping $g\in G$ to $xgx^{-1} \in G$.
Then prove the followings.
(a) The map $\Psi_x$ is a group homomorphism.
(b) The map $\Psi_x=\id$ if and only if $x\in Z(G)$, where $Z(G)$ is the […]

Abelian Normal subgroup, Quotient Group, and Automorphism Group
Let $G$ be a finite group and let $N$ be a normal abelian subgroup of $G$.
Let $\Aut(N)$ be the group of automorphisms of $G$.
Suppose that the orders of groups $G/N$ and $\Aut(N)$ are relatively prime.
Then prove that $N$ is contained in the center of […]

The Index of the Center of a Non-Abelian $p$-Group is Divisible by $p^2$
Let $p$ be a prime number.
Let $G$ be a non-abelian $p$-group.
Show that the index of the center of $G$ is divisible by $p^2$.
Proof.
Suppose the order of the group $G$ is $p^a$, for some $a \in \Z$.
Let $Z(G)$ be the center of $G$. Since $Z(G)$ is a subgroup of $G$, the order […]

Subgroup of Finite Index Contains a Normal Subgroup of Finite Index
Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.
Proof.
The group $G$ acts on the set of left cosets $G/H$ by left multiplication.
Hence […]

The Preimage of a Normal Subgroup Under a Group Homomorphism is Normal
Let $G$ and $G'$ be groups and let $f:G \to G'$ be a group homomorphism.
If $H'$ is a normal subgroup of the group $G'$, then show that $H=f^{-1}(H')$ is a normal subgroup of the group $G$.
Proof.
We prove that $H$ is normal in $G$. (The fact that $H$ is a subgroup […]

Image of a Normal Subgroup Under a Surjective Homomorphism is a Normal Subgroup
Let $f: H \to G$ be a surjective group homomorphism from a group $H$ to a group $G$.
Let $N$ be a normal subgroup of $H$. Show that the image $f(N)$ is normal in $G$.
Proof.
To show that $f(N)$ is normal, we show that $gf(N)g^{-1}=f(N)$ for any $g \in […]

A Group Homomorphism is Injective if and only if Monic
Let $f:G\to G'$ be a group homomorphism. We say that $f$ is monic whenever we have $fg_1=fg_2$, where $g_1:K\to G$ and $g_2:K \to G$ are group homomorphisms for some group $K$, we have $g_1=g_2$.
Then prove that a group homomorphism $f: G \to G'$ is injective if and only if it is […]

## 1 Response

[…] out the post Equivalent definitions of characteristic subgroups. Center is characteristic. for more problems about characteristic […]