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I need to prove that group G of order 36 is solvable. This is what I came with so far:

There is either one 3-sylow subgroup or 4 3-sylow subgroups.

-first case: if there is only one 3-sylow subgroup than P- a 3-sylow subgroup of G is normal. it is also abelian and therefor solvable. [G:P] = 4 and therefor G/P is abelian so it is solvable and we are done.

-second case: if there are four 3-sylow subgroups it means by sylow theory that there is a subgroup N of index 4 so G/N is abelian and therefor solvable and N is of order 9 so it's also abelian and solvable.

the only thing I'm missing is that in the second case I don't know how to prove that N is also normal in G.

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    $\begingroup$ $N$ being normal is inconsistent with the assumption that there are four Sylow subgroups of order $9$ (it's an immediate consequence of Sylow 2). $\endgroup$ – user228113 Sep 18 '17 at 11:12
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In the second case, the trick is as follows: Let $S$ denote the set of all 3-Sylow subgroups, so that $|S|=4$. Let $G$ act on $S$ by conjugation (note that this is a well-defined action). This gives a homomorphism $\varphi : G\to S_4$. Since $|G|>|S_4|$, this map cannot be injective.

So let $N$ be the kernel of this homomorphism. Then $N$ is normal in $G$ and $G/N$ is isomorphic to a subgroup of $S_4$. Hence, $|N|\geq 3$, so I claim that both $N$ and $G/N$ are solvable. Let us consider the cases:

  1. $|N| = 3$, so $|G/N| = 12$. Then $N$ is cyclic and hence solvable. I claim that $G/N$ is also solvable: Consider the number $n_3$ of $3$-Sylow subgroups in $G/N$. If $n_3=1$, then we are done, but if $n_3=4$, then all the 3-Sylow subgroups intersect trivially, so there must be exactly $8=4\times 2$ elements of order 3 in $G/N$. Hence, all the remaining 4 elements must be in a single 2-Sylow subgroup, which must be consequently normal. Hence, $G/N$ is solvable.

  2. $|N|=6$, then $|G/N|=6$. Once again, one of the Sylow subgroups must be normal, so both groups are solvable.

  3. $|N|=9$, then $|G/N|=4$, so both are abelian - hence solvable.

  4. $|N|=12$, then $|G/N| = 3$. Again apply the argument from $1$ to see that both are solvable.

In all cases, both $N$ and $G/N$ are solvable, so $G$ is solvable.

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  • $\begingroup$ why does it follow that N and G/N are solvable? $\endgroup$ – user7080065 Sep 18 '17 at 11:47
  • $\begingroup$ @user7080065: Have added the argument. $\endgroup$ – Prahlad Vaidyanathan Sep 18 '17 at 11:57
  • $\begingroup$ Since all Sylow $3$-subgroups are conjugate, the image of $\phi$ is a transitive subgroup of $S_4$, so $|G/N|$ is divisible by $4$. In fact no Sylow $3$-subgroup can normalize a different Sylow $3$-subgroup, so $|N|$ cannot be divisible by $9$, and hence $|G/N|$ is divisible by $12$. So in fact $|N|=3$ and $|G/N|=12$ is the only case that can occur. $\endgroup$ – Derek Holt Sep 18 '17 at 12:13
  • $\begingroup$ More or less by definition the homomorphism $\phi$ its kernel must be contained in all the Sylow $3$-subgroups. Therefore $|N|=3$ or $|N|=9$. In the latter case $N$ would be a normal Sylow subgroups, so we can rule that out. Ergo, $|N|=3$ is the only case you need to consider. $\endgroup$ – Jyrki Lahtonen Sep 21 '17 at 15:29

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