
Sublattices in Z^2
which of the following are sublattices in Z^2? what is the index??
1.{(x,y) in Z^2 s.t. x+y=1}
2.{(x,y) in Z^2 s.t. x+y=0}
3.{(x,y) in Z^2 s.t. 2x}
4.{(x,y) in Z^2 s.t. 2x2y=0(mod 3)}
5.{(x,y) in Z^2 s.t. 2x2y=0(mod 6)}
A subgroup of Rn is called a (full) lattice in Rn if L = Zv1 +Zv2 + + Zvn for some linearly independent vectors v1, v2 ,..., vn of Rn. For
example, Zn is lattice in Rn.
A sublattice of Zn is a subgroup of finite index.
i dont know how to consider the index . I tried to think about the left coset, but it seems every set can have infinite choices of x and y , many thanks for any hins or suggestion