Advanced High-School Mathematics

218 CHAPTER 4 Abstract Algebra

We have already noted on page 216 that the identity element and
inverses are unique. This says that in denoting the inverse of an ele-
mentg∈Gwe may use, for example, the notationg−^1 to denote this
inverse, knowing that we are unambiguously referring to a unique ele-
ment. However, inverses (and identities) aren’t always denoted in this
way. If we use the symbol + for our binary operation, it’s more cus-
tomary to write “0” for the identity and to write−afor the inverse of
the elementa. Finally it’s worth mentioning that in certain contexts,
the binary operation is simply denoted by “juxtaposition,” writing, for
example xy in place of x∗y. This happens, for instance, in denot-
ing multiplication of complex numbers, polynomials, matrices, and is
even used to denote the binary operation in an abstract group when no
confusion is likely to result.

We shall now survey some very important examples of groups.

1. (The symmetric group) Let X be a set and let (Sym(X),◦)
   be the set of all bijections on X, with function composition as
   the binary operation. At the risk of being redundant, we shall
   carefully show that (Sym(X),◦) is a group.

◦is associative: letσ 1 , σ 2 , σ 3 ∈Sym(X), and letx∈X. Then to

show thatσ 1 ◦(σ 2 ◦σ 3 ) = (σ 1 ◦σ 2 )◦σ 3 we need to show that

they are the same permutations onX, i.e., we must show that

for allx∈X,σ 1 ◦(σ 2 ◦σ 3 )(x) = (σ 1 ◦σ 2 )◦σ 3 (x).But

σ 1 ◦(σ 2 ◦σ 3 )(x) =σ 1 ((σ 2 ◦σ 3 )(x)) =σ 1 (σ 2 (σ 3 (x))),

whereas

(σ 1 ◦σ 2 )◦σ 3 (x) = (σ 1 ◦σ 2 )(σ 3 (x)) =σ 1 (σ 2 (σ 3 (x))),

which is the same thing! Thus we have proved that◦is asso-

ciative.

Existence of identity: Lete:X→Xbe the functione(x) =x, for

allx∈X. Then clearlyeis a permutation, i.e.,e∈Sym(X).