Let k be a nonperfect field of characteristic $2$. Let $a\in k\backslash k^2$, $G=PGL_4(k)$ and and $$H= \left\{ \small\left[\begin{array}{cccc} x & ay &az & aw\\ w & x &ay & az \\ z & w & x & ay \\ y & z & w & x \\ \end{array} \right] \in PGL_4(k) \mid x, y, z, w \in k \right\}.$$ Can anybody find a connected proper overgroup or a nontrivial connected proper subgroup of $H$ in $G$?
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$\begingroup$ Note that $H$ is a group. $\endgroup$– TomoCommented Nov 21, 2014 at 7:35
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2$\begingroup$ Do $w$ and $z$ vary across $k$ too? $\endgroup$– Nick GillCommented Nov 21, 2014 at 14:39
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1$\begingroup$ Being pedantic: I guess the trivial subgroup answers the second part of your question. $\endgroup$– Nick GillCommented Nov 21, 2014 at 17:26
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$\begingroup$ Thanks for comments. The trivial subgroup is not what I want. I am looking for positive dimensional subgroups. Also, I would like to know what happens if k is algebraically closed field of characteristic 2? Are there any over/sub groups of $H$ in $PGL_4(\bar k)$? $\endgroup$– TomoCommented Nov 21, 2014 at 23:40
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$\begingroup$ OK, for a connected subgroup of dimension 1 just take $y=w=0$. The resulting set is is an abelian unipotent subgroup (in particular it has exponent $2$). In fact, if I am not mistaken, over $\overline{k}$ it looks like a root group for $PSp_4(k)$. (Although the question over $\overline{k}$ needs clarification because in this context one cannot choose $a$ to be a non-square.) $\endgroup$– Nick GillCommented Nov 24, 2014 at 17:05
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