Let \(L = \mathbb{Z} \times \mathbb{Z} = \langle (1,0), (0,1) \rangle \subset \mathbb{R}^2\) be the standard integer lattice, and let \(H = \langle (1,0), (0, p) \rangle\) for a positive integer \(p \ge 1\). (1) Prove that \(H\) is a subgroup of \(L\) of index \([L : H] = p\). (2) Describe the quotient group \(L/H\) and the corresponding \(p\)-sheeted covering space of the 2-torus \(T^2 = \mathbb{R}^2 / L\).
Solution
Since \[ H=\langle(1,0),(0,p)\rangle=\mathbb Z\times p\mathbb Z, \] it is visibly a subgroup of \(L=\mathbb Z^2\).
1. Define\[ \phi:L\longrightarrow\mathbb Z/p, \qquad \phi(m,n)=n\pmod p. \]This is surjective and\[ \ker\phi=\mathbb Z\times p\mathbb Z=H. \]Hence the first isomorphism theorem gives\[ L/H\cong\mathbb Z/p. \]Therefore\[ [L:H]=p. \]A set of coset representatives is\[ (0,0),(0,1),\dots,(0,p-1). \]
2. The corresponding connected covering of\[ T^2=\mathbb R^2/L \]is\[ \mathbb R^2/H\longrightarrow\mathbb R^2/L, \qquad [(x,y)]_H\longmapsto[(x,y)]_L. \]Because \([L:H]=p\), this covering has \(p\) sheets.
3. Since\[ \mathbb R^2/H\cong(\mathbb R/\mathbb Z)\times(\mathbb R/p\mathbb Z), \]the covering space is itself a torus. After rescaling the second circle to \(\mathbb R/\mathbb Z\), the covering is equivalent to\[ S^1\times S^1\longrightarrow S^1\times S^1, \qquad (z,w)\longmapsto(z,w^p). \]