Abstract Algebra-The Basic Graduate Year (5).pdf

page 1 of Chapter 5
CHAPTER 5 SOME BASIC TECHNIQUES OF GROUP THEORY
Groups Actingon Sets
In this chapter we are going to analyze and classify groups, and, if possible, break down
complicated groups into ponents. To motivate the topic of this section, let’s
look at the following result.
Cayley’s Theorem Every group is isomorphic to a group of permutations.
Proof. The idea is that each element g in the group G corresponds to a permutation of the
set G itself. If x ∈ G, then the permutation associated with g carries x into = gy,
then premultiplying by g−1 gives x = y. Furthermore, given any h ∈ G, we can solve gx = h
for x. Thus the map x → gx is indeed a permutation of G. The map from g to its associated
permutation is injective, because if gx = hx for all x ∈ G, then (take x =1)g = h. In fact
the map is a homomorphism, since the permutation associated with hg is multiplication by
hg, which is multiplication by g followed by multiplication by h, h ◦ g for short. Thus we
have an embedding of G into the group of all permutations of the set G. ♣
In Cayley’s theorem, a group acts on itself in the sense that each g yields a permutation
of G. We can generalize to the notion of a group acting on an arbitrary set.
Definitions ments The group G acts on the set X if for each g ∈ G there
is a mapping x → gx of X into itself, such that
(1) h(gx)=(hg)x for every g, h ∈ G and x ∈ X
and
(2) 1x = x for every x ∈ X.
As in (), x → gx defines a permutation of X. The main point is that the action of g is
a permutation because it has an inverse, namely the action of g−1. (Explicitly, the inverse
of x → gx is y → g−1y.) Again as in (), the map from g to its associated permutation
Φ(g) is a homomorphism of G into the group SX of permutations of X. But we do not
necessarily have an embedding. If gx = hx for all x, then in () we were able to set
x = 1, the identity element of G, but this resource is not available in general.