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Let $G$ be a finite group. Let $\pi(G)=\{2,3,5\}$ be the set of prime divisors of its order of $G$. If 6 divide the number of Sylow 5-subgroups of $G$ and 10 divide the number of Sylow 3-subgroups of $G$, then whether the group $G$ group with those properties is unsolvable? Thank you so much.

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In particular if the number of Sylow $5$-subgroups of $G$ is 6 and the number of Sylow $3$-subgroups of $G$ is 10, then by the Hall's theorem $G$ is unsolvable group. Because if $G$ is solvable, then $2\equiv 1$ (mod $5$), a contradiction. –  Niki Jun 16 '12 at 13:36
How did you come upon this question? –  S. Carnahan Jun 16 '12 at 15:53
homework 3 exercise 4? –  Anthony Quas Jun 16 '12 at 16:14
@Anthony Quas: I'm sure you do not know what the answer. –  Niki Jun 16 '12 at 19:05
@Niki, I am sure Anthony could work it out easily. Please read mathoverflow.net/faq#whatnot –  Yemon Choi Jun 16 '12 at 21:17
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closed as too localized by Felipe Voloch, Chandan Singh Dalawat, Andres Caicedo, Anthony Quas, Chris Godsil Jun 16 '12 at 17:23

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