Homomorphisms and Isomorphisms

Let $G=(\mathbb{Z},+)$ and $H=(\mathbb{Q}\setminus\{0\},\times)$. Let the map $\phi:G\to H$. Determine whether each of the following rules for $\phi$ makes $\phi$ a homomorphism.

1. $\phi(x)=x$ for all $x\in G$.

2. $\phi(x)=2^{x}$ for all $x\in G$.

Let $G$ and $H$ be groups with identities $e_{G}$ and $e_{H}$ respectively. Show that the trivial homomorphism $\phi:G\to H,g\mapsto e_{H}$ is, indeed, a homomorphism.

Prove 1.7.

Let $G_{1}$ and $G_{2}$ be groups, and let $\phi:G_{1}\to G_{2}$ be a homomorphism. Suppose $H_{1}$ is a normal subgroup of $G_{1}$. Prove or disprove: $\phi(H_{1})\unlhd G_{2}$.

Let $G$ and $H$ be groups, and let $\phi:G\to H$ be a homomorphism. Prove that $|\phi(a)|$ divides $|a|$ for any $a\in G$.

The identity function (or identity map) $\mathrm{id}:S\to S$, where $S$ is a non-empty set, is the function such that $\mathrm{id}(x)=x$ for all $x\in S$.

1. Show that $\mathrm{id}$ is a bijection.

2. If $S$ is a group, show that $\mathrm{id}$ is an isomorphism.

Let $G=(\{1,2,3,4\},\otimes_{5})$ and $H=(\{1,3,7,9\},\otimes_{10})$ be groups.

1. Show that $G=\langle 3\rangle$ and $H=\langle 7\rangle$.

2. Prove that $G\cong H$ by considering the map $\phi:G\to H,3^{k}\mapsto 7^{k}$.

Let $A$, $B$, and $C$ be non-empty sets. Let $f:A\to B$ and $g:B\to C$ be bijections. Let $h=g\circ f$, where $\circ$ represents function composition. That is, $h:A\to C,x\mapsto g(f(x))$.

1. Show that $h$ is a bijection.

2. If $A$, $B$, and $C$ are groups, and $f$ and $g$ are isomorphisms, show that $h$ is an isomorphism.

Let $G$ and $H$ be groups and let the map $\phi:G\to H$ be an isomorphism. Show that if $G$ has a normal subgroup with order $k$, then $H$ also has a normal subgroup of order $k$.

Prove that the “is isomorphic to” relation ($\cong$) is an equivalence relation. In particular, for the groups $G$, $H$, and $K$, prove that

1. $G\cong G$;

2. if $G\cong H$ then $H\cong G$; and

3. if $G\cong H$ and $H\cong K$ then $G\cong K$.

Let the group $G=(\{1,3,7,9\},\otimes_{10})$. Find the positive integer $n$ such that $G\cong\mathbb{Z}_{n}$.

Let $G$ be a group and $g\in G$. Define the map $f:G\to G,x\mapsto gxg^{-1}$. Prove that $f$ is an isomorphism.

Define the map $\phi:\mathbb{Z}\to\mathbb{Z}$ such that $\phi(n)=2n$.

1. Prove that $\phi$ is a homomorphism.

2. Prove that $\phi$ is injective.

3. Prove that there does not exist a homomorphism $\psi:\mathbb{Z}\to\mathbb{Z}$ where $\psi(\phi(n))=n$.

Let $\mathbb{Q}_{>0}$ denote the set of positive rational numbers. Let the groups $G=(\mathbb{Q},+)$ and $H=(\mathbb{Q}_{>0},\times)$. Prove that $G\not\cong H$.

Let $G$ be a group. Define a map $f:G\to G$ such that $f(x)=x^{-1}$ for all $x\in G$. Prove that $G$ is abelian if and only if $f$ is a homomorphism.

Let $G$ and $H$ be groups. Suppose that $\phi:G\to H$ is a surjective homomorphism. Prove that if $G$ is abelian, then so is $H$.

Let $G$ and $H$ be groups. Suppose that $\phi:G\to H$ is a surjective homomorphism. Let $N\unlhd G$. Show that $\phi(N)\unlhd H$. That is, show that the image of $N$ under $\phi$ is a normal subgroup of $H$.

Let the groups $G=\mathbb{Z}_{n}$ and $H=\mathbb{Z}/(n\mathbb{Z})$. Prove that $G\cong H$.

Let $G$ be a group and $N\unlhd G$. Let $B$ be a subgroup of the quotient group $G/N$. Prove that $B=A/N$, where $A$ is a subgroup of $G$ such that $N\subseteq A$.