On mep-relations in the wreath product of groups

Abstract

Let a, b be two long cycles in an alternating group A n , satisfying relations a = [ a , k b ] and b = [ b , k a ] . We show that every pair of elements of the form x = ( X , a ) , y = ( Y , b ) , where the sum of coefficients of X and Y is equal zero, satisfies relations x = [ x , l y ] , y = [ y , l x ] in the wreath product ( S n ≀ Z m ) ′ for m coprime with n and for an l divisible by k. We show also that for n = 5 , 7 , 13 and for m coprime with n, ( S n ≀ Z m ) ′ is generated by such pairs.