Group Theory

Basic Properties of Characteristic Groups

Group Theory Problems and Solutions in Mathematics

Problem 22

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 $\phi \in \Aut(G)$, 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 $G$.

Add to solve later

Proof.

(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 $G$.

Related Question.

Check out the post Equivalent definitions of characteristic subgroups. Center is characteristic. for more problems about characteristic subgroups.

Add to solve later

More from my site

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 […]
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 […]
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 […]
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 […]