Exercise 1.6.16 (Kernels of canonical projections)

Question

Let $A$ and $B$ be groups and let $G$ be their direct product, $A 	imes B$. Prove that the maps $\pi_1 : G 	o A$ and $\pi_2: G 	o B$ defined by $\pi_1((a,b)) = a$ and $\pi_2((a,b)) = b$ are homomorphisms and find their kernels (cf. Exercise 14).

Richard Ganaye · Accepted Answer

\begin{proof} This is the generalization of Exercise 15.

Let $(x,y)$ and $(z,t)$ be any elements of $A 	imes B$. Then
$$\pi_1((x,y) + (z,t))  = \pi_1((x + z, y+ t)) = x+ z =\pi_1((x,y)) + \pi_1(z,t).$$
This shows that $\pi_1$ is a group homomorphism, and similarly $\pi_2$ is a group homomorphism.

Moreover
$$(x,y) \in \ker(\pi_1)\iff \pi_1((x,y)) = 1_A  \iff x = 1_A,$$
so
$$\ker(\pi_1) = \{1_A\} 	imes B,$$
and similarly
$$\ker(\pi_2) = A 	imes \{1_B\}.$$
\end{proof}

Exercise 1.6.16 (Kernels of canonical projections)

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Exercise 1.6.16 (Kernels of canonical projections)

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