# Need Help in Direct Product and Quotient (factor) Group

1. Sep 15, 2010

### Kenji Liew

Hi All. I have several questions on abstract algebra.

Here are my questions and the attempts I had done so far.

(1)Let denote Z as the integer.According to theorem, the direct product of Z3 X Z7 = Z21.
Hence, is Z4 X Z2 is equal to Z8?
Z2X Z2 is equal to Z4?

(2)For Z4 ={0,1,2,3 }, and Z2 ={0,1}.
Then what are the elements will be for Z2 x Z4? Is this a Cartesian product?

(3) We know Z/nZ =Zn, it's a quotient/factor group where nz can be defined as (aH)(bH) = (ab)H. To have a numerical example, we have Z/5Z= Z5.
As usual Z5= {0,1,2,3,4}. But what is the elements contain in Z/5Z and how to find the elements so that the Z/5Z= Z5?

I need some explanations to make me clearer. Thanks a lot ;)

Last edited: Sep 15, 2010
2. Sep 15, 2010

### Eynstone

(3) Just consider addition in Z modulo 5.