The Relation Between the Alternating Group and Standard Row Young’s Diagrams

This study tackles the relationship between the alternating group and young’s diagrams concerning standard rows. The study has been divided into two stages... First, even permutations have been found depending on the conception of partition through formulating an algorithm for this purpose. Secondly, the relationship between the cycle length and partition has been found. Introduction: Let be a non-negative integer, a composition of is a sequence of non-negative integers such that , [5]. For example, if , the following sequences are compositions: . The Relation Between the Alternating Group and Standard ... 91 A composition is said to be a partition for if . In this case of the above mentioned example , the following sequences realize the condition of partition: . young’s diagrams [5] for partition of is : The elements of are called nodes for partition of , and these nodes are elements from , it is represented by a diagram in the form of a system of adjacent square boxes, where of squares are included in the upper row followed by boxes in the row that follows and so on. for example, young diagrams for partition μ in case of is , In case of : , , It is said that the rows of young diagram are standard if the numbers 1 to are included in each row increasingly [7]. For example, the permutations realized in the case of partition (2,1) are only: and [1] If , then is said to be an even (odd) permutation if and only if the multiplication output is: For example, the permutation is even because: 1 2 3 2 3 1 Ilham Matta Yacoob, Hadil H. Sami & Mohammed Kassim Ahmed 92 The alternating group An is defined as the group of all even permutations on a finite group, and it is a subgroup of a symmetric group Sn with elements. [6] The following algorithm has been formulated to find the even cases in Algorithm (1.1) Begin s size of group h 1 N factorial (s) Prod 1 Repeat for x1=[1 to s] { Repeat for x2=[1 to s] a =[x1 x2] if (a has no equal elements) { Repeat for x3=[1 to s] a=[x1 x2 x3] if (a has no equal elements) { Repeat for x4=[1 to s] a=[x1 x2 x3 x4] if (a has no equal elements) { Repeat for x5=[1 to s] { a=[x1 x2 x3 x4 x5] if (a has no equal elements) { Repeat until: a=[x1 x2 ... xs] if (a has no equal elements) { Prod 1 Repeat for c=s to 1 steps(-1) { Repeat for d=c-1 to 1 steps (-1) { } } add a to alt matrix at row h h h+1 } The Relation Between the Alternating Group and Standard ...