Why Read This Book? - Index of

(iii) (Z + 3.8) +Z (Z + 1.2)
(iv) −(Z + 11.23)
(v) (Z + 3.8) −Z (Z + 1.2)
8.3 Quotient Groups 267
(e) Describe addition in R/Z with an analogy similar to the clock arithmetic used
to describe addition in the integers modulo n.
8.3.3 Cosets and Lagrange’s Theorem
Even if G is not an abelian group and the family of cosets of a subgroup of G do
not give rise to a quotient group, we can still derive some important results about
elements and subgroups of G by looking at the cosets of the subgroup. One thing
to keep in mind if G is not abelian is that we have to distinguish between left and
right cosets. For a subgroup H and some fixed g ∈ G, we use the notation
g ∗ H = gH ={g∗ h : h ∈ H} (8.46)
H ∗ g = Hg ={h∗ g : h ∈ H} (8.47)
to denote the left and right coset generated by g, respectively. If G is abelian, then
a particular g will generate the same left and right coset. However, if G is not
abelian, this might not be the case.
EXERCISE 8.3.8 Find all left and right cosets of H ={±1, ±i} in the quaternion
group. For each possible g ∈ Q,isgH = Hg?
EXERCISE 8.3.9 From the group in Exercise 8.1.19, let H ={0, 1}, which is a
subgroup. Evaluate 2 ◦ H and H ◦ 2.
What makes left and right cosets the same for a particular group element is a
question for Section 8.5. Because the results we want to derive here are the same
for either left or right cosets, we will look only at right cosets. First, all cosets of a
given subgroup have the same cardinality.
EXERCISE 8.3.10 If G is a group and H is a subgroup of G, then |H| = |Hg| for
all g ∈ G.
Regardless of whether G is of finite or infinite order, the number of cosets of H
in G might be finite. If so, we call the number of cosets of H the index of H in G,
and we denote this number by (G : H). Naturally, if G is finite, then so is (G : H),
and the following theorem is immediate as an implication of Exercise 8.3.10.
Theorem 8.3.11 (Lagrange). In a finite group, the order of a subgroup divides
the order of the group.