Abstract Algebra: Homomorphisms and Isomorphisms

Let $G$ be the group of matrices $\begin{bmatrix} a & b \\ 0 & c \end{bmatrix}$ where $a$ and $c$ are non-zero real numbers, and $b$ is any real number. Define a homomorphism $F: G \to \mathbb{R}^*$ by $F(\begin{bmatrix} a & b \\ 0 & c \end{bmatrix}) = ac$. Prove that $F$ is a homomorphism and determine its kernel $K$. Using the First Isomorphism Theorem, show that $G/K \cong \mathbb{R}^*$.

Suppose $\phi: R \to S$ is a ring homomorphism. Prove that there exists a unique ring isomorphism $\psi: R/\text{ker}(\phi) \to \text{Im}(\phi)$ such that $\psi(r + \text{ker}(\phi)) = \phi(r)$.

Let $g: \mathbb{Z} \to \mathbb{Z}_5$ such that for any integers $m$ and $n$, $g(mn) = g(m)g(n)$. Does this hold under modular arithmetic?

Consider the groups $G = \mathbb{R}$ under addition and $H = \mathbb{R}^+$ under multiplication. Show that the function $\Phi: G \to H$ defined by $\Phi(x) = e^x$ is a group homomorphism.

Explain how the group U(8) of units modulo 8 (under multiplication modulo 8) is isomorphic to the Klein 4 group $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.

Describe the two isomorphism classes for groups of size four, particularly focusing on $\\\mathbb{Z}/4\\\mathbb{Z}$ and the Klein 4 group, and provide an example group for each.

Determine if S3 is isomorphic to a cyclic group of order 6 and explain why or why not.

Determine if the integers under addition are isomorphic to the even integers under addition, and provide reasoning for your answer.

Consider a homomorphism $\ :$$\ :$from the additive group of all real-valued functions to the set of real numbers, given by the definition \(f) = f(0) . Determine the kernel of this homomorphism.

Given a homomorphism $\phi: \mathbb{Z} \to \mathbb{Z}_8$ defined by $\phi(x) = x \mod 8$, determine if it preserves the homomorphism property.

Given a homomorphism $\phi: GL(2, \mathbb{R}) \to \mathbb{R}^*$ defined by $\phi(A) = \det(A)$, show that it satisfies the homomorphism property.

Find the kernel of the homomorphism $\phi: \mathbb{R}^* \to \mathbb{R}^*$ defined by $\phi(x) = x^2$.

Prove that the mapping from the integers $\mathbb{Z}$ into a ring with unity, where an integer $n$ maps to $n \times 1$, preserves addition and multiplication, and therefore is a homomorphism.