In layperson's terms, what is a general affine group?

Solution 1:

$GA(n,q)$ for $q$ a prime power is the matrix group consisting of matrices in $\mathbb M_{(n+1)×(n+1)}(\mathbb F_q)$ of the form $$\begin{bmatrix}A&v\\\mathbf 0&1\end{bmatrix}$$ where $A$ is an invertible $n×n$ matrix and $v$ is of length $n$. $GA(n,F)$ for $F$ an arbitrary field is much the same, $F$ replacing $\mathbb F_q$.

$GA(1,5)$ may therefore be represented as the group of $5×4=20$ matrices with entries in $\mathbb F_5$ of the form $$\begin{bmatrix}a&b\\0&1\end{bmatrix}$$ with $a\ne0$; that it is a subgroup of $S_5$ holds only in the abstract sense. $GA(2,\mathbb R)$ is famous in SVG as the group of all (invertible) transform strings: $$\text{matrix(a,b,c,d,e,f)}=\begin{bmatrix}a&c&e\\b&d&f\\0&0&1\end{bmatrix}$$ When applied to a 2D point $(x,y,1)^T$, $a=b=c=d=0$ gives a translation, $e=f=0$ gives a linear transformation (rotation, reflection, shearing, etc.) and the general case performs the linear part and then the translation.