Question

Ifis a group that has no proper subgroups, prove that G is a cyclic group of prime order.

Short answer

It has been proved thatG is a cyclic group of prime order.

Step-by-step solution

Prove that order of any element in G is finite

Let $a\in G$such that$a\ne e$.

Then, is a subgroup of G or order greater than 1.

Since G has no proper subgroup, therefore,.

If the order of a is infinite thenis a proper subgroup, which is again not possible.

Hence order of a if finite.

Prove that order of a is prime

Let order of a = n .

If n is not a prime, then n = st for some integers s,t >1.

Then is a subgroup of order s.

So it is a proper subgroup which is not possible.

Hence n must be a prime.

Conclusion

Since . Therefore, order of G is prime.