Pullback Group of Two Group Homomorphisms into a Group

Group Theory Problems and Solutions in Mathematics

Problem 244

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 $G_1 \times G_2$.

 
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Proof.

$M$ is closed under the group operation

Suppose that $(a_1, a_2), (b_1, b_2) \in M$.
By definition, we have
\[f_1(a_1)=f_2(a_2) \text{ and } f_1(b_1)=f_2(b_2). \tag{*}\] The product of $(a_1, a_2)$ and $(b_1, b_2)$ is
\[(a_1, a_2)\cdot (b_1, b_2)=(a_1b_1, a_2b_2).\]

We want to prove that this is in $M$. To see this, note that we have
\begin{align*}
f_1(a_1b_1)&=f_1(a_1)f_1(b_1) && \text{($f_1$ is a homomorphism)}\\
&=f_2(a_2)f_2(b_2) && \text{(by (*))}\\
&=f_2(a_2b_2) && \text{($f_2$ is a homomorphism)}.
\end{align*}

Therefore we have obtained
\[f_1(a_1b_1)=f_2(a_2b_2),\] and the product $(a_1b_1, a_2b_2)$ is in $M$ by definition.

$M$ is closed under inverses

We next prove that if $(a_1, a_2)\in M$ then the inverse
\[(a_1, a_2)^{-1}=(a_1^{-1}, a_2^{-1})\] is also in $M$.

Since $(a_1, a_2)\in M$, we have
\[f_1(a_1)=f_2(a_2). \tag{**}\] Then we have
\begin{align*}
f_1(a_1^{-1})&=(f_1(a_1))^{-1} && \text{($f_1$ is a homomorphism)}\\
&=(f_2(a_2))^{-1} && \text{ (by (**))}\\
&=f_2(a_2^{-1}) && \text{($f_2$ is a homomorphism)}.
\end{align*}
Thus by definition the inverse $(a_1^{-1}, a_2^{-1})$ is in $M$.

Since $M$ is closed under the group operation and inverses, it is a subgroup of $G_1 \times G_2$.

Comment.

In category theory, the subgroup $M$ in the problem is called a pullback of homomorphisms $f_1: G_1 \to H$ and $f_2: G_2 \to H$.


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