Are the following groups isomorphic If so find the isomorphi

Are the following groups isomorphic? If so, find the isomorphism and show that it isis isomorphism. If not explain why they are not. G = (Z_2 times Z_2, +) (Note: the + is mod 2 addition) H (Z_4 +) (Note: the + is mod 4 addition)

Solution

No, this 2 groups are not isomorphic.

If 2 groups are isomorphic then they should preserve some nice properties. Here G and H are not preserving some properties. For example :

H is a cyclic group with 1 as a generator whereas G is not a cyclic group.

H contains element of order 1,2 and 4 whereas G contains elements of order 1 and 2 only, in G there is no element of oreder 4.