An isomorphism from a group $G$ to itself is calledan automorphismof $G$.
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 $\phi \in \Aut(G)$, we have $\phi(H)=H$. In words, this means that each automorphism of $G$ maps $H$ to itself.

Prove the followings.

(a) If $H$ is characteristic in $G$, then $H$ is a normal subgroup of $G$.

(b) If $H$ is the unique subgroup of $G$ of a given order, then $H$ is characteristic in $G$.

(c) Suppose that a subgroup $K$ is characteristic in a group $H$ and $H$ is a normal subgroup of $G$. Then $K$ is a normal subgroup in $G$.

(a) If $H$ is characteristic in $G$, then $H$ is a normal subgroup of $G$.

For each $g\in G$, define a map $\phi_g: G \to G$ defined by $\phi_g(x)=gxg^{-1}$. This is an automorphism of $G$ with the inverse $\phi_{g^{-1}}$.

Since $H$ is characteristic, we have $\phi_g(H)=H$, equivalently we have $gHg^{-1}=H$.
Therefore $H$ is a normal subgroup of $G$.

(b) If $H$ is the unique subgroup of $G$ of a given order, then $H$ is characteristic in $G$.

For any automorphism $\phi \in \Aut(G)$, we have $\phi(H)\subset \phi(G)=G$ and $|H|=|\phi(H)|$. The uniqueness of $H$ implies that $H=\phi(H)$ and thus $H$ is characteristic.

(c) $K$ is characteristic in $H$ and $H$ is normal$G$. Then $K$ is a normal subgroup in $G$.

For each $g \in G$, consider the automorphism $\phi_g$ of $G$ defined in the proof of (a). Since $H \triangleleft G$, we have $\phi_g(H)=H$.
Hence the restriction $\phi_{g}|_{H}$ belongs to $\Aut(H)$.
Now since $K$ is characteristic in $H$, we have $\phi_{g}|_{H}(K)=K$, or equivalently we have $gKg^{-1}=K$ and $K$ is normal in $G$.

Comment.

Let $K$, $H$ be subgroups of $G$. Suppose that $K$ is a normal subgroup of $H$, and $H$ is a normal subgroup of $G$.
In general, we cannot conclude that $K$ is a normal subgroup of $G$.
(For example, consider the dihedral group $D_8$ of order $8$.)

Thus, normality is not transitive.
Part (c) of the problem claims that if in addition, $K$ is characteristic in $H$, then $K$ is normal in $G$.

Equivalent Definitions of Characteristic Subgroups. Center is Characteristic.
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 […]

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 […]

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 […]

Isomorphism Criterion of Semidirect Product of Groups
Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism.
The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation
\[(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),\]
where $a_i […]

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 […]

Normal Subgroups, Isomorphic Quotients, But Not Isomorphic
Let $G$ be a group. Suppose that $H_1, H_2, N_1, N_2$ are all normal subgroup of $G$, $H_1 \lhd N_2$, and $H_2 \lhd N_2$.
Suppose also that $N_1/H_1$ is isomorphic to $N_2/H_2$. Then prove or disprove that $N_1$ is isomorphic to $N_2$.
Proof.
We give a […]

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 […]

A Subgroup of the Smallest Prime Divisor Index of a Group is Normal
Let $G$ be a finite group of order $n$ and suppose that $p$ is the smallest prime number dividing $n$.
Then prove that any subgroup of index $p$ is a normal subgroup of $G$.
Hint.
Consider the action of the group $G$ on the left cosets $G/H$ by left […]

## 1 Response

[…] the post Basic properties of characteristic groups for more problems about characteristic […]