Let H be a subgroup of G, and suppose that k is the smallest positive integer such that g^k is in H. Suppose now that g^l is also in H for some positive integer l. By the extended Euclidean algorithm, m = gcd(k,l) can be expressed as a linear combination ak + bl. But then g^{ak + bl} is an element from H. By the choice of k, we find that k = m, and l is a multiple of k. In particular, g^l is an element of <g^k>. This proves that H = <g^k>.

Let d = gcd(k,n). By the extended Euclidean algorithm there is a relation d = ak + bn. So, for every l we have the relation dl = akl + bnl. This implies that every power of g^d is also a power of g^k. On the other hand, as d divides k, every power of g^k is also a power of g^d. This shows that g^d and g^k generate the same subgroup of G. This proves the second assertion.

This assertion follows immediately from the second one.