Showing two groups are *Not* isomorphic

  • #1
117
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!
 

Answers and Replies

  • #2
117
7
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
Dick
Science Advisor
Homework Helper
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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.
 
Last edited:

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