Let \(G\) be the internal direct product of subgroups \(H\) and \(K\text{.}\) Show that the map \(\phi : G \rightarrow H \times K\) defined by \(\phi(g) = (h,k)\) for \(g =hk\text{,}\) where \(h \in H\) and \(k \in K\text{,}\) is one-to-one and onto.
Let \(G\) and \(H\) be isomorphic groups. If \(G\) has a subgroup of order \(n\text{,}\) prove that \(H\) must also have a subgroup of order \(n\text{.}\)
If \(G \cong \overline{G}\) and \(H \cong \overline{H}\text{,}\) show that \(G \times H \cong \overline{G} \times \overline{H}\text{.}\)
Prove that \(G \times H\) is isomorphic to \(H \times G\text{.}\)
Let \(n_1, \ldots, n_k\) be positive integers. Show that
\begin{equation*} \prod_{i=1}^k {\mathbb Z}_{n_i} \cong {\mathbb Z}_{n_1 \cdots n_k} \end{equation*}if and only if \(\gcd( n_i, n_j) =1\) for \(i \neq j\text{.}\)
Prove that \(A \times B\) is abelian if and only if \(A\) and \(B\) are abelian.
If \(G\) is the internal direct product of \(H_1, H_2, \ldots, H_n\text{,}\) prove that \(G\) is isomorphic to \(\prod_i H_i\text{.}\)
Let \(H_1\) and \(H_2\) be subgroups of \(G_1\) and \(G_2\text{,}\) respectively. Prove that \(H_1 \times H_2\) is a subgroup of \(G_1 \times G_2\text{.}\)
Let \(m, n \in {\mathbb Z}\text{.}\) Prove that \(\langle m,n \rangle = \langle d \rangle\) if and only if \(d = \gcd(m,n)\text{.}\)
Let \(m, n \in {\mathbb Z}\text{.}\) Prove that \(\langle m \rangle \cap \langle n \rangle = \langle l \rangle\) if and only if \(l = \lcm(m,n)\text{.}\)
