G is a group and H is a normal subgroup of G Prove the follo

G is a group and H is a normal subgroup of G. Prove the following:

Suppose that for every x in G, there is an integer n such that xⁿ is in H; then every element of G/H has finite order. Conversely, if every element of G/H has finite order, then for every x in G there is an integer n such that xⁿ is in H.

Solution