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Show that any Group of order 2907 is not a simple group? 2907= 3*3*17*19 I've started with the Sylow 19-subgroup, then the 17-subgroups and finally the 3-subgroups but i couldn't proceed in the proof to find the nontrivial normal subgroup! Please help.

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    $\begingroup$ Can you show your work? $\endgroup$ – Tim Ratigan Nov 27 '13 at 20:11
  • $\begingroup$ @Tim.Ratigan i'm not much familiar with writing using mathematics symbols ! What i did is that I found the number of each sylow subgroup, but couldn't find a way to find a normal one. $\endgroup$ – Enas Nov 27 '13 at 20:24
  • $\begingroup$ This tutorial might help. $\endgroup$ – Tim Ratigan Nov 27 '13 at 20:37
  • $\begingroup$ @Tim.Ratigan thank u :) but i prefer to finish my studying for my midterm exam now, then i'll be ready to learn anything else! $\endgroup$ – Enas Nov 27 '13 at 20:46
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Suppose the $19$-Sylow subgroup(s) isn't(aren't) normal. Are there enough elements left over for the $17$-Sylow subgroup(s) to not be normal? And vice versa.

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  • $\begingroup$ i found that we have 153*18 = 2754 subgroup of order 19 . But then we still have 153 elements ! where is the problem then? $\endgroup$ – Enas Nov 27 '13 at 20:17
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    $\begingroup$ If the $17$-Sylow subgroups aren't normal, how many of them are there? $\endgroup$ – Daniel Fischer Nov 27 '13 at 20:20
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    $\begingroup$ I found a problem while counting the sylow 17-subgroups there number is of the form 17k+1 that divides 9 * 19 so k=0 or k=10 which implies that we have a 171 * 16 = 2736 subgroups of order 17 which contradicts the total number which is 2907 $\endgroup$ – Enas Nov 27 '13 at 20:29
  • $\begingroup$ Right. If the $19$-Sylow subgroups aren't normal, there isn't enough space left for $171$ $17$-Sylow subgroups, so the $17$-Sylow subgroup then must be normal. $\endgroup$ – Daniel Fischer Nov 27 '13 at 20:34
  • $\begingroup$ just simply as this? i mean there is no need to find the form of the normal subgroup such as H1 intersection with H2 for any H1 and H2 sylow subgroups of order 17 for example , or any thing else like Ng( H1 intersection with H2 ) $\endgroup$ – Enas Nov 27 '13 at 20:43

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