SECTION 4.2 Basics of Group Theory 219
Furthermore, for allσ∈Sym(X), and for allx∈X, we have
e◦σ(x) = e(σ(x)) = σ(x), and σ◦e(x) = σ(e(x)) = σ(x),
which proves thate◦σ=σ◦e=σ.
Existence of inverses: Letσ ∈ Sym(X) and let σ−^1 : X → X
denote its inverse function. Thereforeσ−^1 (x) =ymeans pre-
cisely thatσ(y) =xfrom which it follows thatσ−^1 is a permu-
tation (i.e.,σ−^1 ∈Sym(X)) and (σ−^1 ◦σ)(x) =x= (σ◦σ−^1 )(x)
for allx∈X, which says thatσ−^1 ◦σ=e=σ◦σ−^1.I firmly believe that the vast majority of practicing group theorests
consider the symmetric groups the most important of all groups!- (The General Linear Group) LetR be the real number, letn
be a positive integer, and let (GLn(R),·) be the set of alln×n
matrices with coefficients inRand having non-zero determinant,
and where · denotes matrix multiplication. (However, we have
already noted above that we’ll often use juxtaposition to denote
matrix multiplication.) Since matrix multiplication is associative,
since the identity matrix has determinant 1 ( 6 = 0), and since the
inverse of any matrix of non-zero determinant exists (and also has
non-zero determinant) we conclude that (GLn(R),·) is a group.
We remark here that we could substitute the coefficientsRwith
other systems of coefficients, such asCorQ. (We’ll look at another
important example in Exercise 4 on page 223.) - (C,+),(R,+),(Q,+),(Z,+) are all groups.
- Let C∗ = C−{ 0 } (similarly can denote R∗,Q∗,Z∗, etc.); then
(C∗,·) is a group. Likewise, so are (R∗,·) and (Q∗,·) but not
(Z∗,·). - Let Zn denote the integers modulo n. Then (Zn,+) is a group
with identity [0] (again, we’ll often just denote 0); the inverse of
[x] is just [−x]. - Let (G,◦) be the set of automorphisms of some graph. IfXis the
set of vertices of this graph, thenG⊆Sym(X); by the proposition