= Soluble simple group has prime order
For a nontrivial simple <soluble group>, the <commutator subgroup> is either the identity or the whole group. It cannot be the whole group because the <derived series> would never terminate. The group is therefore <Abelian>. Every nonidentity element then generates a normal cyclic subgroup, which must be the whole group. An infinite cyclic group has proper nontrivial subgroups; a finite cyclic group is simple exactly when its order is prime. Thus the soluble simple groups are precisely the cyclic groups of prime order.
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