= Orientation-preserving affine group of the real line
{title2=$\operatorname{Aff}^+(\mathbb R)$}
= Real affine group
{synonym}
The <real affine group> used here is the orientation-preserving subgroup of the <affine group> on the real line. It acts by $x\mapsto e^ax+b$ and has multiplication $(a,b)(a',b')=(a+a',b+e^ab')$. The positive dilation distinguishes this connected component from the full group, which also contains negative dilations.
Here is a small <Lie group> where order matters. Write a point as $(x,1)^T$, and let
$$
D=\begin{pmatrix}1&0\\0&0\end{pmatrix},\quad T=\begin{pmatrix}0&1\\0&0\end{pmatrix},\quad S_a=\exp(aD)=\begin{pmatrix}e^a&0\\0&1\end{pmatrix},\quad U_b=\exp(bT)=\begin{pmatrix}1&b\\0&1\end{pmatrix}.
$$
The <one-parameter subgroups> $S_a$ and $U_b$ give <dilations> and <translations>. With <matrix multiplication> acting rightmost first, $S_aU_b$ sends $x$ to $e^ax+e^ab$, whereas $U_bS_a$ sends $x$ to $e^ax+b$. At $a=\log2$ and $b=1$, translate then double gives $2x+2$, while double then translate gives $2x+1$. Starting at $x=1$ gives endpoints $4$ and $3$.
The same order dependence survives infinitesimally: the <matrix commutator> is $[D,T]=DT-TD=T$. Indeed, the <group commutator> satisfies
$$
S_aU_bS_{-a}U_{-b}=U_{(e^a-1)b}.
$$
At $a=b=0.1$, the residual <translation> is $0.01051709$, close to the leading product $ab=0.01$. It measures a second-order failure of the four moves to cancel. This connects finite composition directly to the <Lie bracket> on the <Lie algebra>; see https://www.math.toronto.edu/mein/teaching/LectureNotes/lie.pdf[Meinrenken, Section 4, Proposition 4.7].
\Image[/lie-affine-noncommutation.png]
{title=Scaling and translation have different endpoints when their order is reversed}
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