Products of quotient groups

#group_theory

Suppose $G$ is a nontrivial finite group and $H, K ⊴ G$ are normal subgroups with $gcd (| H |, | K |) = 1$ .
(a) Define a nontrivial group homomorphism $ϕ : G \to G / H \times G / K$ .
(b) Prove $G$ is isomorphic to a subgroup of $G / H \times G / K$ .
(c) Suppose $gcd (m, n) = 1$ . Prove $Z_{m n} ≅ Z_{m} \times Z_{n}$ .

View

L A T E X

code

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}