Prove that every group of order 58 is not simple. So I know that 58 = 2 ⋅ 29. I assume G is simple. I'm having trouble using the Sylow Theorems to show that this is not Simple. In particular, computing the number of sylow groups and using that to show the group isn't Simple
2026-03-25 07:40:30.1774424430
Help proving Sylow's Theorem order 58
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Let $P \in Syl_{29}(G)$. Such a $P$ exists by Sylow's (First) Theorem (or just by Cauchy's Theorem). Then $|G:P|=2$ and it is basic group theory knowledge that subgroups of index $2$ are normal. So $G$ cannot be simple.