Why Read This Book? - Index of

268 Chapter 8 Groups
Proof. Let G be a group, and let H be a subgroup of G. Since all cosets of H are
disjoint and have the same cardinality as H, we have that
|G| = (G : H) × |H|
If x is an element of a finite group G, then Lagrange’s Theorem says that the
order of the subgroup generated by x divides the order of G. Since the cardinality
of (x) is the same as the order of the element x (Theorem 8.2.21), we have that
o(x) | o(G). Beginning here, it is possible to prove several results about elements
of a group.
EXERCISE 8.3.12 Suppose o(G) = n, and g ∈ G. Then g n = e. 11
EXERCISE 8.3.13 Suppose G is cyclic of order n and k is a positive integer.
If k | n, then there exists a subgroup of G of order k.
EXERCISE 8.3.14 A group of prime order is cyclic. 12
8.4 Permutation Groups
In this section, we take an in-depth look at a particular group and two particularly
important subgroups that derive from it. Our main purpose is to become familiar
with an especially important non-abelian group. Then in Section 8.5, we will use
this group as motivation for the definition of a normal subgroup. Requiring a
subgroup to be normal is just the right thing to patch up the hole we left in our
derivation of quotient groups by requiring that the group be abelian.
8.4.1 Permutation Groups Defined
Let A be a non-empty set, and let S be the set of all bijections from A to itself. We
want to take a close look at the group formed on S with the operation of composition.
First, note that composition is well defined and closed on S by Exercise 8.1.4,
so that S with the operation of composition has properties G1 and G2.
The fact that composition is associative (G3) is a mere exercise in the
manipulation of parentheses. For if we pick any a ∈ A, then
[(f ◦ g) ◦ h](a) = (f ◦ g)[h(a)] =f(g(h(a))) = f [(g ◦ h)(a)] =[f ◦ (g ◦ h)](a)
(8.48)
Thus [(f ◦ g) ◦ h](a) =[f ◦ (g ◦ h)](a) for all a ∈ A, so that (f ◦ g) ◦ h =
f ◦ (g ◦ h), and ◦ is associative on S. Writing i : A → A, the identity function,
11 Apply Theorem 8.2.21 and Lagrange’s Theorem to (g).
12 Any element except e is a generator.