Products of quotient groups

Mar 17, 2025 4:17 PM

Suppose $G$ is a nontrivial finite group and $H, K ⊴ G$ are normal subgroups with $gcd (| H |, | K |) = 1$ .

Define a nontrivial group homomorphism $ϕ : G \to G / H \times G / K$
Prove $G$ is isomorphic to a subgroup of $G / H \times G / K$ .
Suppose $gcd (m, n) = 1$ . Prove $Z_{m n} ≅ Z_{m} \times Z_{n}$ .

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Suppose $G$ is a nontrivial finite group and $H,K\mathrel{\unlhd}G$ are normal subgroups with $\gcd(|H|,|K|)=1$.
\begin{enumerate}[label=\alph*)]
	\item Define a nontrivial group homomorphism $\phi:G\to G/H\times G/K$
	\item Prove $G$ is isomorphic to a subgroup of $G/H\times G/K$.
	\item Suppose $\gcd(m,n)=1$. Prove ${\bf Z}_{mn}\cong {\bf Z}_m\times {\bf Z}_n$.
\end{enumerate}