Let $\mathbb {K}$ be a field. Let $f: \mathbb {K}^2 \rightarrow \mathbb {K}^2; x \mapsto Ax+b$ be an affine transformation. Suppose $f$ has a fixed point line (i.e. a line such that every point on that line is a fixed point of $f$). When does the linear map $x \mapsto Ax$ have a fixed point line?
- What I tried:
I tried to construct a fixed point line of the linear map from the one of $f$, but to no avail. I know that $(0,0)$ is a fixed point of the linear map. If I could obtain one other fixed point I would be done, since by linearity the line through the origin and that point would consist only of fixed points. So it boils down to finding a fixed point of the linear map other than the origin. Another thought: By our assumption the coefficient matrix of the inhomogeneous system of linear equations $(A-I_2)x=-b$ has rank one. Now we are interested in the homogeneous system. Any hints?
It would be nice if you also considered the case when any subgroup of $G$ of the index $p$ is abelian. The straight lines are defined as $l:\,tx+v$, where $x,v\in K^2$ are fixed vectors and $t$ runs through all possible values of $K$. By convention, there exists a line for which $$ A(tx+v)+b=tx+v, \forall t\in K. $$ Then $$ t(A-I)x=-b-(A-I)v, \forall t\in K. $$ This is only possible if $(A-I)x=0$ or $Ax=x$. And then $A(tx)=tx$. That's it.