Definition:Affine Group of One Dimension

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Let $S$ be the set of mappings $f_{a b}: \R \to \R$ defined as:

$S := \left\{{f_{a b}: x \mapsto a x + b: a \in \R_{\ne 0}, b \in \R}\right\}$

The algebraic structure $\left({S, \circ}\right)$, where $\circ$ denotes composition of mappings, is called the $1$-dimensional affine group on $\R$ and can be denoted $\operatorname{Af}_1 \left({\R}\right)$.

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