Why Read This Book? - Index of

266 Chapter 8 Groups
identity element H ∗ e = H, and inverses (H ∗ a) −1 = H ∗ a −1 is called the
quotient group of G created by modding out the subgroup H. Elements of G/H
are called right cosets of H.
For equivalence classes in general, we know that [a] =[b] if and only if a and b
are equivalent; otherwise [a] and [b] are disjoint. In the context of Definition 8.3.5,
these facts become H ∗ a = H ∗ b if and only if a ∗ b −1 ∈ H; otherwise H ∗ a and
H ∗ b are disjoint. In other words, a and b generate the same coset of H if and only
if a ∗ b −1 ∈ H. Otherwise the cosets they generate are disjoint.
Example 8.3.6 Let G be the set of all functions f : R → R. For two functions f
and g, we define their sum f + g by the rule [f + g](x) = f(x) + g(x). Clearly, the
function 0 defined by 0(x) = 0 for all x is the identity element, and −f is the function
defined by [−f ](x) =−f(x). Thus (G, +, 0, −) is an abelian group. Let H be
the subset of G consisting of all constant functions, which is clearly a subgroup of G.
What do the elements of the quotient group G/H look like? If f is a given
function in G, then H + f ={h + f : h ∈ H} is the set of all translations of f up
and down in the xy-plane by the constant functions in H. That is, g ∈ H + f if and
only if there exists some constant function h such that g = h + f . Equivalently,
g ∈ H + f , provided g − f ∈ H, which means that g − f is a constant function.
This might remind you of a fact from calculus. If f is any antiderivative of a function
f1, then H + f is the set of all the antiderivatives of f1. If you like, every coset
can be addressed by choosing a representative element, perhaps the function in
the coset that passes through the origin. �
EXERCISE 8.3.7 Since the real numbers under addition form an abelian group
with the integers as a subgroup, we may discuss R/Z.
(a) In constructing R/Z, what is the definition of equivalence that gives rise to
the right cosets of Z?
(b) Construct the following cosets by listing some of their elements.
(i) Z + 1.4
(ii) Z + √ 2
(iii) Z + 5
(c) What is a convenient subset of the real numbers from which we may choose
a unique representative element of each coset? 10
(d) Evaluate the following in R/Z. Express your answer as a coset Z + r, where
r is an element of your answer to part (c).
(i) (Z + 2.2) +Z (Z + 8.14)
(ii) (Z + 3.8) +Z (Z + 0)
10 There are infinitely many answers, but one of them is considered standard.