# Showing two groups are *Not* isomorphic

Tags:
1. Oct 2, 2015

### DeldotB

1. The problem statement, all variables and given/known data
Good day,

I need to show:

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

2. Relevant equations

None

3. 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?

2. Oct 2, 2015

### 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?

3. Oct 2, 2015

### Dick

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

Last edited: Oct 2, 2015