Permutation groups

A bunch whose elements are permutations and in which the product of two permutations is often a permutation whose effect is equivalent to the successive application of the first two.

A one – one mapping f of a finite non-empty set S onto itself is called a permutationIf the set S consists of n distinct elements, then a one-one mapping of S onto itself is called a permutation of degree n.

Let S = {a1,a2,a3,.........ana1,a2,a3,………an)

Then we denote a permutation f on the set S in a two-rowed notation.

f = [a1,b1,a2,b2,..........,..........,ai,bi,............,............,anbn][a1,a2,……….,ai,…………,anb1,b2,……….,bi,…………,bn]

So that in the first row all the elements of S are written in a certain order and

f(a11) = b11

f(a22) = b22

f(a33) = b33….

f(aii) = bii ………..

f(ann) = bnn

where bii‘s are aii‘s.

Note : Interchange of columns does not change the permutation. (12233441)(12342341) = (23123441)(21343241) = (12413423)(14322143)

Types of permutations

Some of the types of permutations are given below.

1)  Equality of permutations:

Two permutations f and g of a set S are said to be equal, if f(a) = g(a),  a  S.

Eg : If f =  (132132)(123312) and g = (213213)(231123)

are two permutation of degree 3, then we have

f = g since, f(1) = g(1) = 3, f(2) = g(2) = 1, f(3) = g(3) = 2

(After interchanging columns)

2) Identity permutation:

A permutation on the set S is called the identity permutation, if it maps each element of S onto itself. It is usually denoted by the symbol I.

Thus I(a) = a,  a  S

Eg : (112233....................nn)(123………n123………..n)

is the identity permutation of degree n,

3) Inverse permutations:

Since, a pemutation is a one-one onto mapping and hence, it is invertible, i.e., every permutation f on a set P = (a1,a2,a3,..........an1,a2,a3,……….an)  has a unique inverse permutation denoted by f1−1

Eg: If f = (a1b1a2b2....................anbn)(a1a2………anb1b2………..bn)

Then, f1−1 = (b1a1b2a2....................bnan)(b1b2………bna1a2………..an)

Cyclic Permutation (cycle)

A permutation f on a set S is called a cyclic permutation of length l, if there exist x11, x22, x33,

……….., xll  S such that f(x11) = x22, f(x22) = x33, ……., f(xl1l−1) = xll,

f(xll) = x11 and f(x) = x,  x  S, if x  S x11, x22,……………., xll

Example: (122331)(123231) is a cyclic permutation of length 3.

(1223344551)(1234523451) is a cyclic permutation of length 5.

We denote the cyclic permutation (x1x2x2x3.....................xlx1)(x1x2………xlx2x3…………x1) by the symbols (x11, x22, …………., xll)

Thus, cyclic permutation (122334455661)(123456234561) is expressed as (1 2 3 4 5 6).

Symmetric group

Let A(S) denote the set of all permutation on a non-empty set S.

Then, A(S) form a group under the composition of maps. Moreover, if S contain n elements, then the

permutation group A(S) contains n! elements. The group of n! permutations of a set of n elements is called symmetric group (Snn) of degree n.

Example 1: Write down all the elements of the permutation group(Symmetric group) S33 on three elements 1, 2 and 3.

Solution: Let S = {1, 2, 3}

Then, there are 3! = 6 elements in S33.

S33 = (112233)(123123) (122133)(123213) (132231)(123321) (112332)(123132)(122331)(123231) (132132)(123312)

which can be written as

S33 = {1, (1 2), (1 3), (2 3), (1 2 3), (1 3 2)}

Author: Deepak Chahar

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