Homomorphism of Upper Triangular Matrices to Nonzero Reals

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 $G/K \cong \mathbb{R}^*$.

In this problem, we explore concepts centered around homomorphisms within group theory, specifically dealing with matrices. The group G in question consists of upper triangular matrices where the diagonal elements are non-zero real numbers. By defining a homomorphism from this group to the multiplicative group of non-zero real numbers, we dive into examining the properties and kernel of this mapping, which is foundational in understanding the internal structure of groups and their relationships with other groups.

The key to solving this problem is to confirm the homomorphic nature of the defined mapping and identify the kernel. The kernel here serves as the set of elements in G that are mapped to the identity in the codomain. Understanding kernels is crucial as they provide significant insights into the structure and characteristics of group homomorphisms, essentially indicating which information is 'lost' during the mapping process.

Further, utilizing the First Isomorphism Theorem, we see how the quotient of the group G by this kernel can be shown to be isomorphic to the multiplicative group of non-zero real numbers. This theorem is a powerful tool in abstract algebra, shedding light on how groups can be decomposed into simpler, understandable pieces. Hence, this problem elegantly combines elements of homomorphisms, kernels, and quotient groups to deepen our grasp of algebraic structures in a high-level, interconnected manner.