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.© BrainMass Inc. brainmass.com December 15, 2022, 7:31 pm ad1c9bdddf
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 ...
This provides examples of proofs regarding prime subgroups.