Advanced High-School Mathematics

SECTION 4.2 Basics of Group Theory 213

Note that the above example depends heavily on the fact thatZ

is closed under both addition and multiplication.)

• Let GLn(R)⊆Matn(R) denote the matrices of determinant 6 = 0,
  and let GL+n(R)⊆Matn(R) denote the matrices of positive deter-
  minant. Then both of these sets are closed under multiplication;
  neither of these sets are closed under addition.


• The subset { 0 ,±i,±j,±k} ⊆ Vect 3 (R) is closed under vector
  cross product×. The subset{ 0 ,i,j,k}⊆Vect 3 (R) is not. (Why
  not?)


• The subset{− 1 , 0 , 1 }⊆Zis closed under multiplication but not
  under addition.


• Let X be a set and let Sym(X) denote the set of permutations.
  Fix an elementx∈X and let Symx(X)⊆Sym(S) be the subset
  of all permutations which fix the elements. That is to say,

Symx(X) = {σ∈Sym(X)|σ(x) =x}.

Then Symx(X) is closed under function composition◦(Exercise 5).

We have two more extremely important binary operations, namely

addition and subtraction on Zn, the integers modulo n. These

operations are defined by setting

[a] + [b] = [a+b], and [a]·[b] = [a·b], a, b∈Z.^8

We shall sometimes drop the [·] notation; as long as the context is

clear, this shouldn’t cause any confusion.

(^8) A somewhat subtle issue related to this “definition” is whether it makes sense. The problem is
that the same equivalence class can have many names: for example if we are considering congruence
modulo 5, we have [3] = [8] = [−2], and so on. Likewise [4] = [−1] = [14]. Note that [3] + [4] = [7] =
[2]. Since [3] = [−2] and since [4] = [14], adding [−2] and [14] should give the same result. But they
do: [−2] + [14] = [12] = [2]. Here how a proof that this definition of addition really makes sense (i.e.,
that it iswell defined) would run. Let [a] = [a′] and [b] = [b′]. Thena′=a+ 5kfor some integer
kandb′=b+ 5lfor some integerl. Therefore [a′] + [b′] = [a+ 5k] + [b+ 5l] = [a+b+ 5k+ 5l] =
[a+b] = [a] + [b]. Similar comments show that multiplication likewise makes sense. Finally this
generalizes immediately toZn, for any positive integern.