Direct Products of Groups

Prove 1.2.

Simplify $(s,rs)(r^{2}s,r^{3})$ in $D_{3}\times D_{4}$.

Find all positive integers $m$ and $n$, where $1<m<n$, such that $\mathbb{Z}_{m}\times\mathbb{Z}_{n}\cong\mathbb{Z}_{180}$.

Let $G=\{1,5\}$ and $H=\{1,7\}$ be groups under $\otimes_{12}$. Find the internal direct product of $G$ and $H$.

Let $\mathcal{S}=\{1,5,7,11\}$, $G=\{1,5\}$, and $H=\{1,7\}$ be groups under $\otimes_{12}$. Find the value of $n$ such that $\mathcal{S}\cong(\mathbb{Z}_{n})^{2}$.

Let $G$ and $H$ be abelian groups. Prove that $G\times H$ is also an abelian group.

Let $G$ and $H$ be groups. Prove that $G\times H\cong H\times G$.

Let $G=\mathbb{Z}_{6}$, and consider the subgroups $H=\{0,2,4\}$ and $K=\{0,3\}$ of $G$. Determine whether $G$ is the internal direct product of $H$ and $K$, justifying your answer.

Consider the Klein four-group $\mathrm{V}$ with presentation

Show that $\mathrm{V}\cong(\mathbb{Z}_{2})^{2}$.