Group Direct Product/Examples/R-0 x R/Isomorphism to Set of Affine Mappings on Real Line under Composition

Theorem

Let $G$ be the Cartesian product of $\R \setminus \set 0$ with $\R$:

$G = \set {\tuple {a, b} \in \R^2: a \ne 0}$

Let $\circ$ be the group product on $G$ defined as:

$\tuple {a_1, b_1} \circ \tuple {a_2, b_2} = \tuple {a_1 a_2, a_1 b_2 + b_1}$

Let $S$ be the set of all real functions $f: \R \to \R$ of the form:

$\forall x \in \R: \map f x = r x + s$

where $r \in \R_{\ne 0}$ and $s \in \R$

Let $\struct {S, \circ}$ be the algebraic structure formed from $S$ and the composition operation $\circ$.

Then $\struct {G, \circ}$ is (group) isomorphic to $\struct {S, \circ}$.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{N}$