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How can I show that a group with 380 elements is not simple?

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Since you are new, I want to give you some advice about the site: To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people are much more willing to help you if you show that you've tried the problem yourself. – Zev Chonoles Jul 9 '12 at 0:40
In fact, groups of order $p(p+1)$, for $p$ a prime, always have a normal subgroup, of order $p$ or order $p+1$. – user641 Jul 9 '12 at 21:03

Note that $380=2^2\times 5 \times 19$. A Sylow subgroup associated to $19$ is necessarily cyclic of order $19$; if it is normal, we are done. And if it is not normal, then there must be twenty such subgroups; any two intersect trivially, since they are groups of prime order, so the twenty subgroups account for $20\times 18 + 1 = 361$ elements.

Now, consider the Sylow $5$-subgroups; how many can there be if there are twenty Sylow $19$-subgroups?

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