Group Theory

Abelian Groups and Surjective Group Homomorphism

Problem 167

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 $G’$.

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Definitions.

Recall the relevant definitions.

A group homomorphism $f:G\to G’$ is a map from $G$ to $G’$ satisfying
\[f(xy)=f(x)f(y)\] for any $x, y \in G$.
A map $f:G \to G’$ is called surjective if for any $a\in G’$, there exists $x\in G$ such that
\[f(x)=a.\]
A surjective group homomorphism is a group homomorphism which is surjective.

Proof.

Let $a, b\in G’$ be arbitrary two elements in $G’$. Our goal is to show that $ab=ba$.
Since the group homomorphism $f$ is surjective, there exists $x, y \in G$ such that
\[ f(x)=a, f(y)=b.\]

Now we have
\begin{align*}
ab&=f(x) f(y)\\
&=f(xy) \text{ since } f \text{ is a group homomorphism}\\
&=f(yx) \text{ since } G \text{ is an abelian group}\\
&=f(y)f(x) \text{ since } f \text{ is a group homomorphism}\\
&=ba.
\end{align*}
Therefore, we obtain $ab=ba$ for any two elements in $G’$, thus $G’$ is an abelian group.

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