So it is cyclic: it there were two primes dividing the order of the group then either Cauchy's Teorem or Sylow's theorems
would give us at least two different proper non-trivial subgroups, each of order a different prime, so the order of the
group is divisible only by one prime, say p. It can't be that the order is p, since then there is no proper non-trivial sbgp's at
all, and it can't be , since then there are at least 2 different proper non-trivial sbgp's, of
orders (why??) , so...