
index
What does it mean by if n>0, then the index of <n> in Z (integers) is the number of left cosets of <n>?
i understand that it is the number of congruence class modulo n but i cant seem to apply it to an example.
for instance,
in Z (mod 13), there are a total of 4 left cosets, namely
[2]H, [4]H, [3]H and [7]H such that all elements in G appears only once.
but by the theorem appear, it seems to me that there should be 13 cosets since Z is in modulo 13..
is there something wrong with my interpretation of this definition?
thanks!

If what you call $\displaystyle <n>$ is the subgroup $\displaystyle n\mathbb Z$, then you are correct in saying that this subgroup of $\displaystyle \mathbb Z$ has $\displaystyle n$ cosets in $\displaystyle \mathbb Z$.
I deduce from your post that what you call $\displaystyle H$ is the subgroup $\displaystyle 13\mathbb Z$. Why would it have only 4 cosets? It has 13, namely $\displaystyle H,2H,3H,\dots,12H$. They form a partition of $\displaystyle \mathbb Z$.

i thought it was 4 cosets as there are 4 left cosets
[2]H={[2],[g],[5]}
[4]H={[4],[10],12]}
[3]H={[1],[3],[9]}
and [7]H={[7],[8],[11]} as i thought that when the questions talks about cosets, they mean either left or right cosets..
or is that not the case?
beside thinking of them as left and right cosets, is there another way i can think of them cos im quite confused with the idea of cosets.
from what you said, it seems to me that cosets are similar to congruence class in Z13..