Abstract Algebra: Theory and Applications

Prove that $det(AB)=det(A)det(B)$ for $A,B\in G{L}_2(\mathbb{R})\text{.}$ This shows that the determinant is a homomorphism from $G{L}_2(\mathbb{R})$ to ${\mathbb{R}}^{\ast }\text{.}$

Let $A$ be an $m\times n$ matrix. Show that matrix multiplication, $x\mapsto Ax\text{,}$ defines a homomorphism $\varphi :{\mathbb{R}}^n\to {\mathbb{R}}^m\text{.}$

Let $\varphi :\mathbb{Z}\to \mathbb{Z}$ be given by $\varphi (n)=7n\text{.}$ Prove that $\varphi$ is a group homomorphism. Find the kernel and the image of $\varphi \text{.}$

Describe all of the homomorphisms from ${\mathbb{Z}}_{24}$ to ${\mathbb{Z}}_{18}\text{.}$

Describe all of the homomorphisms from $\mathbb{Z}$ to ${\mathbb{Z}}_{12}\text{.}$

If $G$ is an abelian group and $n\in \mathbb{N}\text{,}$ show that $\varphi :G\to G$ defined by $g\mapsto g^n$ is a group homomorphism.

If $\varphi :G\to H$ is a group homomorphism and $G$ is abelian, prove that $\varphi (G)$ is also abelian.

If $\varphi :G\to H$ is a group homomorphism and $G$ is cyclic, prove that $\varphi (G)$ is also cyclic.

Show that a homomorphism defined on a cyclic group is completely determined by its action on the generator of the group.

If a group $G$ has exactly one subgroup $H$ of order $k\text{,}$ prove that $H$ is normal in $G\text{.}$

Prove or disprove: $\mathbb{Q}/\mathbb{Z}\cong \mathbb{Q}\text{.}$

Let $G$ be a finite group and $N$ a normal subgroup of $G\text{.}$ If $H$ is a subgroup of $G/N\text{,}$ prove that ${\varphi }^{-1}(H)$ is a subgroup in $G$ of order $|H|\cdot |N|\text{,}$ where $\varphi :G\to G/N$ is the canonical homomorphism.

Let $G_1$ and $G_2$ be groups, and let $H_1$ and $H_2$ be normal subgroups of $G_1$ and $G_2$ respectively. Let $\varphi :G_1\to G_2$ be a homomorphism. Show that $\varphi$ induces a homomorphism $\bar{\varphi }:(G_1/H_1)\to (G_2/H_2)$ if $\varphi (H_1)\subset H_2\text{.}$

If $H$ and $K$ are normal subgroups of $G$ and $H\cap K=\{e\}\text{,}$ prove that $G$ is isomorphic to a subgroup of $G/H\times G/K\text{.}$

Let $\varphi :G_1\to G_2$ be a surjective group homomorphism. Let $H_1$ be a normal subgroup of $G_1$ and suppose that $\varphi (H_1)=H_2\text{.}$ Prove or disprove that $G_1/H_1\cong G_2/H_2\text{.}$

Let $\varphi :G\to H$ be a group homomorphism. Show that $\varphi$ is one-to-one if and only if ${\varphi }^{-1}(e)=\{e\}\text{.}$

Given a homomorphism $\varphi :G\to H$ define a relation $\sim$ on $G$ by $a\sim b$ if $\varphi (a)=\varphi (b)$ for $a,b\in G\text{.}$ Show this relation is an equivalence relation and describe the equivalence classes.