Mathematical Foundation of Computer Science

DHARM

ALGEBRAIC STRUCTURES 85

p 1 =

123

231

132

213

312

(^23123)
F
H
I
K =
F
H
I
K =
F
H
I
,,ppK
and p 4 =
321
132
213
321
231
(^56312)
F
H
I
K =
F
H
I
K =
F
H
I
,,ppK
Let p be the permutation where,
p =
aaa
bbb
12 3
123
F
HG
I
KJ
then, inverse of permutation is denoted by p–1 and is defined as,
p–1 =
bbb
aaa
123
12 3
F
HG
I
KJ
Similarly, a permutation is said to be an identity permutation if image of every element
of set X is same to the corresponding element, i.e., if X = {a, b, c} then p(a) = a, p(b) = b, and
p(c) = c, then permutation p where,
p = FHabcabcIK is called an identity permutation.

4.7 Order of a Group................................................................................................................

Let (X, ) be a group, then order of the X is denoted by O(X) is the number of elements in
group X. For example if group X consisting of m elements then O(X) = m. If X is infinite then
O(X) = ∞.
Let x ∈ X then order of an element x, denoted as O(x) is n if an = e (identity element).
For example, consider X = {1, – 1, i, – i} is a group under usual multiplication operation × i.e.
(X, ×).
Then order of the group is, i.e., O(X) = 4. The order of its elements is determined as
follows :
l O(1) = 1 [∵ 11 = 1 (identity element)]
l O(– 1) = 2 [∵ (– 1)^2 = 1 (identity element)]
l O(i) = 4 [∵ (i)^4 = 1 (identity element)]
l O(– i) = 4 [∵ (– i)^4 = 1 (identity element)]
Example 4.6. Let X = {1, ω, ω^2 } where ω is cube root of unity, is a group (X, +). Determine the
order of the group X and order of its elements.
Sol. Since, set X contains three elements hence, O(X) = 3. The order of the elements is deter-
mined as follows :
From the operation table shown in Fig. 4.5 the group (X, ×) has an identity element 1
Therefore,
l O(1) = 1 [∵ 11 = 1 (identity element)]
l O(ω) = 3 [∵ (ω)^3 = 1 (identity element)]
l O(ω^2 ) = 3 [∵ (ω^2 )^3 = 1 (identity element)]