Modern algebra chapter 6

reference: Modern Algebra An Introduction (John R. Durbin)

 

CHAPTER 6

 

<Permutations>

Sn=set{1,2,3,4∙∙∙,n}

Sn can be represented in two-row form

 

menas 1↦2, 2↦4, 3↦3, 4↦1

 

<identity>

 

<inverse>

 

<non-Abelian>

β∙α ≠ α∙β

 

<identity can omitted>

 

<composition>

 

Begin with 1 (1

The cycle (3 4) fixes 1, and then (1 2 4) gives 1↦2, so write (1 2

The cycle (3 4) fixes 2, and then (1 2 4) gives 2↦4, so write (1 2 4

The cycle (3 4) gives 4↦3, and then (1 2 4) fixes 3, so write (1 2 4 3

The cycle (3 4) gives 3↦4, and then (1 2 4) gives 4↦1,

so 3↦1 by the product and we close the cycle, giving (1 2 4 3).

<disjoint>

if αβ = βα then, disjoint

if αβ ≠ βα then, not disjoint

<cyclic decomposition>