Some notes regarding Permutation Groups

Permutation Groups

Preliminaries

A permutation can be represented as a function on a set of points and can also be represented in a cyclic notation. We normally do not write out cycles of length one (these are also called fixed points). So if [\pi = (0, 3 , 4, 1)(2)(5, 6)(7)] then the fixed points are 2 and 7 and we rewrite it as [\pi = (0, 3 , 4, 1)(5, 6)]

Multiplication of two permutations, say, (\alpha) and (\beta) are defined by function composition that is ((\alpha)(\beta))(x) = (\alpha)((\beta) x)

Thus we can see that composition of two permutations is again a permutation.

Some basic algorithms

Algorithm 1

This is used to multiply permutations

MULT(n, (\alpha), (\beta), (\gamma))

for i (\Leftarrow) 0 to n-1 do (\pi_{0})[i] (\Leftarrow) (\alpha)[(\beta)[i]] for i (\Leftarrow) 0 to n-1 do (\gamma)[i] (\Leftarrow) (\pi_{0})[i]

Note that we use an auxiliary permutation here to prevent bugs arising because of modifications to the arrays (\alpha) & (\beta)

Algorithm 2

This is used to compute the inversion of a permutation

INV(n, (\alpha),)

for i (\Leftarrow) 0 to n-1 do (\beta)[(\alpha)[i]] (\Leftarrow) i

Algorithm 3

This is used to convert a cycle to an array

CYCLE_TO_ARRAY (n, C)

for i (\Leftarrow) 0 to n-1 do (A)[i]] (\Leftarrow) i

i (\Leftarrow) i Set l to be the length of string C while i < l: if C[i] = "(" then: i (\Leftarrow) i + 1 if C[i] (\in) {0, 1, ..., 9} then: Get x starting at i z (\Leftarrow) y

Increment i to the position after x