How many different non-isomorphic Abelian groups of order 4 are there?

A. 2
B. 3
C. 4
D. 5

Question

How many different non-isomorphic Abelian groups of order 4 are there?

A. 2
B. 3
C. 4
D. 5

Accepted Answer

A. 2 By the Fundamental Theorem of Finitely Generated Abelian Groups, every finite Abelian group is isomorphic to a direct product of cyclic groups of prime power order. For order 4 = 2^2, the possible decompositions are Z_4 (cyclic, one element of order 4) and Z_2 x Z_2 (Klein four-group, every non-identity element has order 2). These two are not isomorphic to each other, giving exactly 2 distinct Abelian groups of order 4.

How many different non-isomorphic Abelian groups of order 4 are there? A. 2 B. 3 C. 4 D. 5