Thread: Steiner-Triple Systems

Hey
There was no combinatorics section so this is probably the best section to post this question in...

I was looking at how to construct STS(n) where p=1 mod 6 and p prime.
The following method I know works but I was just wondering why it works... if someone could explain that would be great:

Find w such that w^3=1 mod p
Construct the set {1,w,w^2} which are the roots of positive unity...
We can form a group {1,w,w^2,-1,-w,-(w^2)} with multiplication under mod p.
This is the 6 roots of unity.

Once we have done that we can find all the right cosets until we have exhausted the integers mod p. If p=6m+1 then we will have m cosets... Now we take the first three elements of each set equivalent to 1,w and w^2 and make a new set. We will have m sets each with 3 elements.


Property) Now, each non-zero element x of the integers mod p will have a unique solution u-v=x where u and v are in the same set that we have constructed. So if out sets of 3 are B1,B2,....BM we can construct the STS(p) be taking the set {B1+k,B2+k,...,BM+k:k is in integers mod p} which I get because of the property I explained. But how do we know constructing sets in this way will give this property?

I hope I have explained it well enough.
Thanks

Does anyone know?