Well, I'm not sure what you're missing without seeing your working out but firstly are you correctly finding the subgroup?
<(1,0> = {(1,0), (0,0)}
So I am having a problem. The question is:
Find the left coset of <(1,0)> in Z_{2 }x Z_{4}.
Since there are 8 elements in Z_{2 }x Z_{42 }and only 2 in (1,0), there should be 4 left cosets, but I can only find one.. (1,0). What am I missing?
Then, the process is:
Take a particular element in Z_{2} x Z_{4} and add it to each of the elements in <(1,0)>. Do this for each element in Z_{2} x Z_{4}. This will give you 8 cosets, but as you say you should find there are only 4 DISJOINT cosets each with two elements.
Hi cloeannx3,
You're correct that there should be 4 left/right cosets - nice job! I'll work out two cosets and leave the other two for you.
Let's start by writing out the following:
<(1,0)> =
To determine what the (left) cosets are we go down the list of elements in making cosets of <(1,0)>. The first two cosets are
1)
2) Notice that this (left) coset is exactly the same as the coset
Notice that the sets two cosets we have found in 1 and 2 are disjoint.
Does this help get things on the right track? Let me know if anything is unclear. Good luck!