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Prime simple subgroups

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a) Show that if |G|= pq, where p and q are prime, then G is not simple.

b) Show that the only simple groups of order less than 36 are of prime order.

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This provides examples of proofs regarding prime subgroups.

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(a)

If p = q, then G is abelian, and contains a subgroup of order p (by Cauchy), which is necessarily normal.

If p =/= q, say p < q, then there is a unique Sylow q-subgroup Q, which must be normal.

( That is, if n_q denotes the number of Sylow q-subgroups, then n_q must divide p, and be congruent to 1 (mod q). )

But if n_q = 1 + kq with k > 0, then 1 + kq > q > p, so cannot ...

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