Definition:Affine Group of One Dimension
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Definition
Let $S$ be the set of mappings $f_{a b}: \R \to \R$ defined as:
- $S := \set {f_{a b}: x \mapsto a x + b: a \in \R_{\ne 0}, b \in \R}$
The algebraic structure $\struct {S, \circ}$, where $\circ$ denotes composition of mappings, is called the $1$-dimensional affine group on $\R$ and can be denoted $\map {\operatorname {Af}_1 } \R$.
Also see
Sources
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Exercise $(2)$