The general linear group $GL(2,3) $ of $2\times 2$ matrices over modulo $3$ does not have normal Sylow $2$-subgroup as it has three Sylow $2$-subgroups. I have to find the maximal normal $2$- subgroup of $GL(2,3)$. According to me it has normal $2$- subgroup of order $8$, one can see https://groupprops.subwiki.org/wiki/2-core_of_general_linear_group:GL(2,3). Can I say it is maximal normal $2$- subgroup. Please suggest me . Thanks .

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    $\begingroup$ If it has a normal subgroup $N$ of order $8$, and its Sylow $2$-subgroup of order $16$ is not normal, then $N$ must be its largest normal $2$-subgroup. BTW, it has three Sylow $2$-subgroups. $\endgroup$ – Derek Holt Jan 28 at 8:13
  • $\begingroup$ @DerekHolt I am also saying that it’s Sylow subgroups are not normal $\endgroup$ – neelkanth Jan 28 at 8:27
  • $\begingroup$ Yes it’s has three Sylow subgroups ... I will edit ... $\endgroup$ – neelkanth Jan 28 at 8:29
  • $\begingroup$ @DerekHolt so my argument is correct ? $\endgroup$ – neelkanth Jan 28 at 8:31
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    $\begingroup$ If it had two normal subgroups of order eight, $N_1$ and $N_2$. Then $N_1N_2$ would be a subgroup. Furthermore, it would be normal. And its order would be $8^2/|N_1\cap N_2|$. Implying that $N_1N_2$ would have to be a Sylow 2-subgroup. But that wasn't normal, so... $\endgroup$ – Jyrki Lahtonen Jan 28 at 8:57

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