Elementary Abstract Algebra- Examples and Applications, 2019a

19.3 GROUP ACTIONS ASSOCIATED WITH SUBGROUPS AND COSETS807
19.3.1 The integer lattice
In this section we’ll take a close look at a group action on a set of cosets. This
example can be thought of as a two-dimensional version of Example 19.3.9,
and can be envisioned using computer graphics.
Let G be the xy-plane under addition (that is, G =(R 2 , +)). Let H =
Z × Z, which is a subgroup of G (H is called the integer lattice: see
Figure 19.3.1). Cosets of H in G may be written as a + H = {(x + m, y +
n) :m, n ∈ Z}, wherea := (x, y) can be any element of G. Recall from
Proposition 15.2.4 that cosets form a partition, so H and its cosets partition
R 2 .
Figure 19.3.1. Diagram showing H (the integer lattice) with the unit
square shaded. This figure and the similar figures in the section were created
using the software “GeoGebra” (see http://www.geogebra.org).
The unit square (the shaded area in Figure 19.3.1) is the area on the
xy-plane that is [0, 1) × [0, 1), meaning the square includes the points on
the x and y axes, but not on the lines x = 1 and y = 1. Note that H
has only one point in the unit square, namely (0, 0), similarly any coset of
the form a + H has only one point in the unit square (we will prove this