Eckmann–Hilton Argument: Group Operation is a Group Homomorphism

Group Theory Problems and Solutions in Mathematics

Problem 268

Let $G$ be a group with the identity element $e$ and suppose that we have a group homomorphism $\phi$ from the direct product $G \times G$ to $G$ satisfying
\[\phi(e, g)=g \text{ and } \phi(g, e)=g, \tag{*}\] for any $g\in G$.

Let $\mu: G\times G \to G$ be a map defined by
\[\mu(g, h)=gh.\] (That is, $\mu$ is the group operation on $G$.)

Then prove that $\phi=\mu$.
Also prove that the group $G$ is abelian.

 
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Proof.

$\phi=\mu$

Since $\phi$ is a group homomorphism, for any $g, g’ \in G$ and $h, h’ \in H$, we have
\begin{align*}
\phi( (g,h)(g’,h’))=\phi(g,h)\phi(g’,h’).
\end{align*}
The left hand side is equal to
\[\phi(gg’, hh’),\] and thus we have
\[\phi(gg’, hh’)=\phi(g,h)\phi(g’,h’).\] Setting $g’=h=e$, we have
\begin{align*}
\phi(g, h’)&=\phi(g, e)\phi(e, h’)\\
&= gh’ && \text{ by (*)}\\
&=\mu(g, h’).
\end{align*}
Since this equality holds for any $g \in G$ and $h’\in H$, we obtain
\[\phi=\mu\] as required.

$G$ is an abelian group

Now we prove that $G$ is an abelian group.
Let $g, h \in G$ be any two elements.
Then we have
\begin{align*}
gh&=\phi(e,g)\phi(h,e) && \text{ by (*)}\\
&=\phi(eh, ge) && \text{ since $\phi$ is a homomorphism}\\
&=\phi(h,g)\\
&=\mu(h,g)=hg.
\end{align*}
Thus we have proved that $gh=hg$ for any $g, h \in G$. Thus the group $G$ is abelian.

Combined version

Here is a combined version of the proofs of the two claims at once.
We have for any $g, h\in G$,
\begin{align*}
&\mu(g,h)\\
&=gh=\phi(e,g)\phi(h,e) && \text{ by (*)}\\
&=\phi((e,g)(h,e)) && \text{ since $\phi$ is a homomorphism}\\
&=\phi(eh, ge)\\
&=\phi(h,g) \\
&=\phi(he,eg)=\phi((h,e)(e,g))\\
&=\phi(h,e)\phi(e,g) && \text{ since $\phi$ is a homomorphism}\\
&=hg \qquad \text{ by (*)}\\
&=\mu(h,g)
\end{align*}

From these equalities, we see that $\phi=\mu$ and $G$ is an abelian group.

Remark.

This argument is called the Eckmann–Hilton argument.


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