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Take $T=\big(\left< (123) \right> \times \left< (456) \right> \times \left< (789) \right>\big) \rtimes \left< (147)(258)(369) \right> \leq S_9$.

Does there exist an injective group homomorphism from $T$ into $\operatorname{GL}(9,3)$, such that its image is closed under the operation $(A,B)\mapsto A+B-I$?

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    $\begingroup$ Could you possibly provide some kind of motivation for this question? $\endgroup$
    – Derek Holt
    Commented Oct 22, 2020 at 7:33
  • $\begingroup$ Actually it started with $C_3 \wr C_3$, it can be embedded in $S_9$, as the set $T$ taken above. If I will be able to find this kind of mapping into $GL(9,3)$, then I can say many things about the group ring $F_3(C_3 \wr C_3)$ . $\endgroup$
    – HIMANSHU
    Commented Oct 22, 2020 at 7:42
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    $\begingroup$ We can assume that the image lies in a given Sylow $3$-subgroup of ${\rm GL}(9,3)$, so you are asking whether there is a subgroup $S$ of the group of upper unitriangular matrices in ${\rm GL}(9,3)$ with $S \cong C_3 \wr C_3$, such that $\{ g - I : g \in S \}$ is a group under addition. $\endgroup$
    – Derek Holt
    Commented Oct 22, 2020 at 7:53
  • $\begingroup$ @DerekHolt Yes, exactly. $\endgroup$
    – HIMANSHU
    Commented Oct 22, 2020 at 7:56
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    $\begingroup$ @GeoffRobinson That's not true - $C_3 \wr C_3$ embeds into ${\rm GL}(4,3)$. But I have checked by computer that no embedding into ${\rm GL}(n,3)$ for $n=4$ or 5 has the required property. I may be able to check $n=6$ - I am not sure yet - but I will not be able to get up to $n=9$ by naive computation. Still, I am starting to think that no such embedding is possible. $\endgroup$
    – Derek Holt
    Commented Oct 23, 2020 at 7:45

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