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I'm having some trouble with the following exercise: I am asked to determine how many Sylow $p$-subgroups the group $G$ might have, where $|G|=240=2^4\cdot3\cdot5$. I am not sure how to interpret this "might": I mean, I applied the third Sylow Theorem and computed the possible values for $n_p$, the number of $p$-Sylows in $G$, but it seems I'm not able to go beyond this and give a sharper estimate. Can someone help please?

pozio
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  • Look at some examples of similar kind questions,for example this:http://math.stackexchange.com/questions/352501/a-group-of-order-30-has-a-normal-5-sylow-subgroup?rq=1 – Arpit Kansal Sep 08 '15 at 08:11
  • A p-sylow subgroup any subgroup of order $p^{\alpha}$ where $\alpha$ is the maximum power of $p$ that divides $o(G)$. In the case of the group of order 240, you have to look for a 2-Sylow,3-Sylow and 5-Sylow subgroups. – Miz Sep 08 '15 at 08:13
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    I would guess that they just want you to determine all possible $n_p \equiv 1 \bmod p$ with $n_p$ dividing $|G|$, for $p=2,3$ and $5$. – Derek Holt Sep 08 '15 at 08:44
  • @DerekHolt that's exactly what I did and I get a rather long list: $n_2=1,3,5,15$, $n_3=1,4,10,16,40$ and $n_5=1,6,16$. My problem is that I don't know how to say more, for example whether one of these $p$-Sylows is normal. – pozio Sep 08 '15 at 16:56
  • Perhaps you are not expected to say more? In fact I think all of these values are possible except for $n_3=10$ and $n_3=40$, but proving that would be moderately difficult, It is not true that there has to be a normal Sylow $p$-subgroup for some $p$. (consider $S_5 \times C_2$). – Derek Holt Sep 09 '15 at 08:05

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