O'Reilly logo

Basic Abstract Algebra by S. K. Jain, S. R. Nagpaul, P. B. Bhattacharya

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

CHAPTER 7

Permutation groups

1 Cyclic decomposition

Let us recall that a cycle σ = (a1ar) in Sn is a permutation such that σ(ai) = ai + 1 for i = 1,…, r – 1, σ(ar) = a1, and σ(x) = x for every other x in n. Two cycles (a1,…,ar) (b1,bs) in Sn are disjoint permutations if and only if the sets {a1,…, ar} and {b1,…,bs} are disjoint. Note that a cycle of length r can be written in r ways, namely, as (a1 … ar) and (aiai+1ara1ai– 1), i = 2,…, r. A cycle of length r is also called an r-cycle.

1.1 Theorem. Any permutation images is a product of pairwise disjoint cycles. This cyclic factorization is unique except for the order in which the cycles ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required