Showing two groups are *Not* isomorphic

[DeldotB]

Homework Statement

Good day,

I need to show:

$$\mathbb{Z}_{4}\oplus \mathbb{Z}_{4}$$is not isomorphic to $$\mathbb{Z}_{4}\oplus \mathbb{Z}_{2}\oplus \mathbb{Z}_{2}$$

Homework Equations

None

The Attempt at a Solution

I was given the hint that to look at the elements of order 4 in a group. I know $$\mathbb{Z}_{4}\oplus \mathbb{Z}_{4}$$ will have the elements: (0,0)(0,1)(0,2)(0,3)(1,0)(1,1)...(3,3).

Im a little confused on how to find the order of say (1,2) in $$\mathbb{Z}_{4}\oplus \mathbb{Z}_{4}$$.
I know how to find the order of say <3> in $$\mathbb{Z}_{4}$$ (order=4/gcd(3,4)=4) but how can I do it with the direct sum elements?

Thanks in advance!

 

[DeldotB]

Can anyone tell me this is correct:

In Z4 direct sum Z4, say we look at the element (2,0)
Since <2> has order 2 in Z4 and <0> has order one in Z4, it follows that (2,0) has order 2 because the lcm(2,1) is 2?

 

[Dick]

Can anyone tell me this is correct:

In Z4 direct sum Z4, say we look at the element (2,0)
Since <2> has order 2 in Z4 and <0> has order one in Z4, it follows that (2,0) has order 2 because the lcm(2,1) is 2?

Correct. The conclusion is clearly correct since (2,0)+(2,0)=(0,0). Try some others until you can see why that rule works.