# A Group Homomorphism and an Abelian Group

## Problem 207

Let $G$ be a group. Define a map $f:G \to G$ by sending each element $g \in G$ to its inverse $g^{-1} \in G$.
Show that $G$ is an abelian group if and only if the map $f: G\to G$ is a group homomorphism.

## Proof.

### $(\implies)$ If $G$ is an abelian group, then $f$ is a homomorphism.

Suppose that $G$ is an abelian group. Then we have for any $g, h \in G$
\begin{align*}
f(gh)&=(gh)^{-1}=h^{-1}g^{-1}\\
&=g^{-1}h^{-1} \text{ since } G \text{ is abelian}\\
&=f(g)f(h).
\end{align*}
This implies that the map $f$ is a group homomorphism.

### $(\impliedby)$ If $f$ is a homomorphism, then $G$ is an abelian group.

Now we suppose that the map $f: G \to G$ is a group homomorphism.
Then for any $g, h \in G$, we have
$f(gh)=f(g)f(h) \tag{*}$ since $f$ is a group homomorphism.
The left hand side of (*) is
$f(gh)=(gh)^{-1}=h^{-1}g^{-1}.$ Thus we obtain from (*) that
$h^{-1}g^{-1}=g^{-1}h^{-1}.$ Taking the inverse of both sides, we have
$gh=hg$ for any $g, h \in G$.
It follows that $G$ is an abelian group.

## Related Question.

Another problem about the relation between an abelian group and a group homomorphism is:
A group is abelian if and only if squaring is a group homomorphism

### More from my site

• Group Homomorphism Sends the Inverse Element to the Inverse Element Let $G, G'$ be groups. Let $\phi:G\to G'$ be a group homomorphism. Then prove that for any element $g\in G$, we have $\phi(g^{-1})=\phi(g)^{-1}.$     Definition (Group homomorphism). A map $\phi:G\to G'$ is called a group homomorphism […]
• Pullback Group of Two Group Homomorphisms into a Group Let $G_1, G_1$, and $H$ be groups. Let $f_1: G_1 \to H$ and $f_2: G_2 \to H$ be group homomorphisms. Define the subset $M$ of $G_1 \times G_2$ to be $M=\{(a_1, a_2) \in G_1\times G_2 \mid f_1(a_1)=f_2(a_2)\}.$ Prove that $M$ is a subgroup of $G_1 \times G_2$.   […]
• A Group is Abelian if and only if Squaring is a Group Homomorphism Let $G$ be a group and define a map $f:G\to G$ by $f(a)=a^2$ for each $a\in G$. Then prove that $G$ is an abelian group if and only if the map $f$ is a group homomorphism.   Proof. $(\implies)$ If $G$ is an abelian group, then $f$ is a homomorphism. Suppose that […]
• 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 Groups and Surjective Group Homomorphism Let $G, G'$ be groups. Suppose that we have a surjective group homomorphism $f:G\to G'$. Show that if $G$ is an abelian group, then so is $G'$.   Definitions. Recall the relevant definitions. A group homomorphism $f:G\to G'$ is a map from $G$ to $G'$ […]
• A Homomorphism from the Additive Group of Integers to Itself Let $\Z$ be the additive group of integers. Let $f: \Z \to \Z$ be a group homomorphism. Then show that there exists an integer $a$ such that $f(n)=an$ for any integer $n$.   Hint. Let us first recall the definition of a group homomorphism. A group homomorphism from a […]
• 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 […]
• Surjective Group Homomorphism to $\Z$ and Direct Product of Abelian Groups Let $G$ be an abelian group and let $f: G\to \Z$ be a surjective group homomorphism. Prove that we have an isomorphism of groups: $G \cong \ker(f)\times \Z.$   Proof. Since $f:G\to \Z$ is surjective, there exists an element $a\in G$ such […]

### 1 Response

1. 03/05/2017

[…] Another problem about the relation between an abelian group and a group homomorphism is: A group homomorphism and an abelian group. […]

##### Order of the Product of Two Elements in an Abelian Group

Let $G$ be an abelian group with the identity element $1$. Let $a, b$ be elements of $G$ with order...

Close