Further Properties of Homomorphisms

Consider the map $\phi:\mathbb{Z}_{3}\to\mathbb{Z}_{6},n\mapsto 2n$. Determine whether $\phi$ is a homomorphism and, if so, find its image.

Let $i$ be the imaginary unit, i.e. $i^{2}=-1$. Let $G=(\mathbb{Z},+)$ and $H=(\langle i\rangle,\times)$. Let the map $\phi:G\to H,n\mapsto i^{n}$. Show that $\phi$ is a homomorphism and hence find $\ker\phi$.

Let $G$ and $H$ be groups. Prove that a homomorphism $\phi:G\to H$ is injective if and only if $\ker\phi$ is trivial, i.e. $\ker\phi=\{e_{G}\}$.

Let $\phi:G\to H$ be a homomorphism between finite groups $G$ and $H$. Prove that $|G|=|\mathrm{im}\;\phi|\times|\ker\phi|$.

Let $G$ be a finite group. Let $H$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$. Prove that

Suppose $x$ and $y$ are positive integers such that $y=mx$ for some integer $m$. Let $H=x\mathbb{Z}$ and $N=y\mathbb{Z}$ be groups under addition.

1. Explain why $N\subseteq H$.

2. Find a group $G$ such that both $H$ and $N$ are normal subgroups of $G$.

3. Hence find the order of $H/N$.

Let $G$ be a group. Prove that $G/G$ is isomorphic to the trivial group.

Let the group $R=(\mathbb{R},+)$. Also let the groups $G=R^{2}$ and $H=\left\{(r\times\sqrt{2},r\times\sqrt{3})\mid r\in R\right\}$ be under component-wise addition. Prove that $G/H\cong R$.

Let $G$ be a group. Let $H$ and $K$ be subgroups of $G$ such that $K\subseteq H$. Prove that $HK=H$.

Let $G$ be an abelian group under the operation $\ast$. Let $I=\{(g,g^{-1})\mid g\in G\}$ be a group under component-wise application of $\ast$.

1. Show that $I\cong G$.

2. Prove that $G^{2}/I\cong G$.

Let $G=m\mathbb{Z}$ and $H=n\mathbb{Z}$ be groups under addition, where $m\mid n$ and $m\neq n$. Let the map $\phi:G\to\mathbb{Z}/({\frac{n}{m}}\mathbb{Z})$ be defined such that

Prove that $G/H\cong\mathbb{Z}_{\frac{n}{m}}$.