Prove that Z4 is a group with respect to 0Solutionthe standa

Prove that Z4 is a group with respect to 0.

Solution

the standard proof goes something like this:

(0,1) in Z2 x Z2 has order 2:

(0,1) + (0,1) = (0,0) (since 1+1 = 2 = 0 (mod 2)).

(1,0) in Z2 x Z2 also has order 2:

(1,0) + (1,0) = (0,0).

now if Z2 x Z2 and Z4 were isomorphic, Z4 would contain (at least) 2 elements of order 2 (the images of (1,0) and (0,1) under the isomorphism).

but:

0 is of order 1

1+1 = 2
1+1+1 = 3
1+1+1+1 = 0 <---1 is of order 4

2+2 = 0 <---2 is of order 2

3+3 = 2 (since 6 = 2 (mod 4))
3+3+3 = 1 (since 9 = 1 (mod 4))
3+3+3+3 = 0 <---3 is of order 4.

since Z4 has only one element of order 2 (namely, 2), it cannot be isomorphic to Z2 x Z2.