= Elementary abelian subgroups in GL3 over F2
{title2=$K_L=\{I+vf:f(v)=0\},\quad K_W=\{I+wf:w\in\ker f\}$}
Every involution of $GL_3(2)$ is a <transvection> $I+vf$, because its nilpotent part has square zero and rank one. Two distinct commuting transvections either have common image line or common fixed plane. This gives two types of elementary abelian groups of order four, displayed above. No larger elementary abelian $2$-subgroup is possible: a transvection commuting with all three elements of a common-image group must have that same image, and the dual argument applies to a common-fixed-plane group. Their <normalizers> are respectively point and plane stabilizers of order $24$. The common fixed-space dimensions one and two distinguish their conjugacy classes.
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