# Algebra: A Computational Introduction by John Scherk By John Scherk

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Extra resources for Algebra: A Computational Introduction

Example text

N − 1}. For example, in Z/2Z there are two congruence classes: 0 + 2Z = {2s | s ∈ Z}, the even integers, and 1 + 2Z = {1 + 2s | s ∈ Z}, the odd integers. We can define addition on Z/nZ by a ¯ + ¯b = a + b . 2 this makes sense. You can think of this as adding two natural numbers a and b in {0, 1, . . , their sum is the remainder after division of a + b by n. This addition in Z/nZ is associative and commutative: (¯ a + ¯b) + c¯ = a ¯ + (¯b + c¯) a ¯ + ¯b = ¯b + a ¯. And a ¯ + ¯0 = ¯0 a ¯ + (−a) = ¯0 .

N}. Cycles are particularly simple. We shall show that any permutation can be written as a product of cycles, in fact there is even a simple algorithm which does this. Let's first carry it out in an example. Take ( ) 1 2 3 4 5 6 7 8 . α = 2 4 5 1 3 8 6 7 We begin by looking at α(1), α2 (1), . . We have α(1) = 2, α(2) = 4, α(4) = 1. As our first cycle α1 then, we take the 3-cycle ( ) 1 2 3 4 5 6 7 8 α1 = . 2 4 3 1 5 6 7 8 28 CHAPTER 2. PERMUTATIONS The smallest number which does not occur in this cycle is 3.

M' can be used to make useful calculations in permutation groups. m; Next you have to know how our notation for permutations is implemented. In this package, a permutation (in mapping notation) is given by the list of its images with the header, M. 4. b ( ) 1 2 3 4 5 6 Out= 6 4 1 5 2 3 To find the inverse of a, enter In:= Inverse[a] ( ) 1 2 3 4 5 6 Out= 3 1 5 2 6 4 A permutation in cycle notation is written as the list of its cycles, which are in turn lists, and is preceded by the header P.