I assume that you mean no non-trivial proper subgroups. The answer is yes but in the spirit of your question...no. If is a finite group and where then a simple application of Cauchy's theorem says that must have an element of order . More generally, Sylow's first theorem implies that if is a finite group and with prime then must contain a subgroup of order . From this it's easy to see that the only groups you mention have order or where is prime.