- #1

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- 7

## Homework Statement

Good day,

I need to show:

[tex]\mathbb{Z}_{4}\oplus \mathbb{Z}_{4} [/tex]is not isomorphic to [tex]\mathbb{Z}_{4}\oplus \mathbb{Z}_{2}\oplus \mathbb{Z}_{2}[/tex]

## 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 [tex]\mathbb{Z}_{4}\oplus \mathbb{Z}_{4} [/tex] 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 [tex]\mathbb{Z}_{4}\oplus \mathbb{Z}_{4} [/tex].

I know how to find the order of say <3> in [tex]\mathbb{Z}_{4}[/tex] (order=4/gcd(3,4)=4) but how can I do it with the direct sum elements?

Thanks in advance!