Today , We Shall see the most Important Finite Non abelian group from " Cryptography " point of view. 1. Let P , Q R , S be any given distinct positive integers greater than or equal to " 2 " 2. DEFINE N = P.Q.R.S + 1 . THEN ( P , N ) = ( Q , N ) = ( R , N ) = ( S , N ) = 1 3. G = { ( a , b , c , d , e ) | a , b , c , d , e BELONGS TO { 0 ,1 , 2 , 3 , . . . N - 1 } } Now in " G " WE DEFINE A BINARY OPERATION AS FOLLOWS : 4. ( a , b , c , d , e) * ( f , g , h , i , k ) = ( x , y , z , w , l ) [ a ,b ,c,d,e ,f ,g ,h i , k belongs to { 0 ,1,2,3, . . . , N - 1 } ] where x = a + f ( Mod N ) y = b + g ( Mod N ) z = c + h ( Mod N ) w = d + i ( Mod N ) l = f b + g c + h d + e + k ( Mod N ) Now " G " is Non abelian group of order = N.N.N.N.N LET US WORK on this Non abelian group " G " Then the map " T " from G to G ( a , b , c , d , e ) ------------> ( a , b , c , d ,e) * ( a , b , c , d ,e ) * .... * ( a , b ,c ,d ,e ) [ " P " times ] is a Permutation on " G " ( MULTIPLICATION BY * APPEARS " P " TIMES ) Similarly , we can replace " P " BY Q , R , S [ N = P .Q .R.S + 1 ]