## ABSTRACT

A group whose elements are permutations on a given finite set of objects (symbols) is called a permutation group on these objects. Let a1, a2, . . . , an denote n distinct objects, and let b1, b2, . . . , bn be any arrangement of the same n objects. The operation of replacing each ai by bi, 1 ≤ i ≤ n, is called a permutation performed on the n objects. It is denoted by

S :

( a1 a2 · · · an b1 b2 · · · bn

) (B.1)

and is called a permutation of degree n. If the symbols on both lines in (B.1) are the same, the permutation is called the identical permutation and is denoted by I. Let

T :

( b1 b2 · · · bn c1 c2 · · · cn

) , U :

( a1 a2 · · · an c1 c2 · · · cn

) ,

then U is the product of S and T , i.e., U = ST , since

U = ST =

( a1 a2 · · · an b1 b2 · · · bn

)( b1 b2 · · · bn c1 c2 · · · cn

) =

( a1 a2 · · · an c1 c2 · · · cn

) .