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The group $\mathrm{GL}(n,\mathbb{Z})$ acts on $(\mathbb{Z}/2\mathbb{Z})^n$ by right multiplication (the same kind of things can be done with left action). I denote by $H\subset \mathrm{GL}(n,\mathbb{Z})$ the isotropy group of $(1,\dots,1)$, which corresponds to the group of matrices such that each column has an odd number of odd values.

Then $H$ has index $2^n-1$ is maximal and if $n\ge 3$, it is generated by permutations, diagonal matrices and two simple upper triangular matrices with two ones or one two outside of the diagonal.

I came to this group with questions of algebraic geometry and my questions are the following:

Is this group studied somewhere in the literature? Is this useful to something else? Are the (easy) results about generators stated above written somewehere?

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  • $\begingroup$ But for $n=2$ it is still a subgroup of finite index in the finitely generated subgroup ${\rm GL}(2,{\mathbb Z})$, so it must be finitely generated. $\endgroup$
    – Derek Holt
    Mar 14 '14 at 8:49
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Your action is really an action of ${\rm GL}(n,\mathbb{Z}/2\mathbb{Z}).$ Your subgroup is the full pre-image of the stabilizer of a particular $1$-dimensional subspace. The stabilizers of $1$-dimensional subspaces are maximal parabolic subgroups ( of the finite group), and all these (1-dimensional subspace stabilizers) are conjugate. The field with two elements is a bit different because there is no scalar action to worry about, so vector stabilizers coincide with the stabilizers of the corresponding one dimensional subspace. The maximal parabolics of ${\rm GL}(n,\mathbb{Z}/2\mathbb{Z})$ containing the natural Borel subgroup are generated by the Borel subgroup and $n-2$ natural generating reflections of the Weyl group. Your subspace stabilizer (viewed in the finite image) lies over a different Borel subgroup, but the conjugating matrix is clear.

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  • $\begingroup$ Thanks for the answer. However, it seems to me that it is not exactly what I wanted. I agree that my group is a full pre-image of a natural subgroup of $\mathbb{GL}(n,\mathbb{F}_2)$ but I am more interested in the subgroup of $\mathbb{GL}(n,\mathbb{Z})$ itself, as this one appears to be a group acting geometrically on my space (and not the other one). Is this group studied somewhere? $\endgroup$ Mar 14 '14 at 9:08

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