If G is a finite group of order n, then there is at least one element of G that has order n. If G is a finite group of order n, and n is prime, then there is at least one element of G that has order n. Prove ONE of the following theorems from the text. Every subgroup of a cyclic group is cyclic. If G is a cyclic group of order n, then G is isomorphic to Z n .
Every group is isomorphic to a group of permutations.
Write sigma as a product of disjoint cycles. Compute the inverse of sigma. Let G be a group. I. Any subgroup of a cyclic group is cyclic. If G is a cyclic group of order n, then G is isomorphic to Z n .