We shall give an introduction to the problem area concerning the well known Duval's conjecture, which was announced to be solved in .
Let W= W(xx,...,Xj)bc any word in thej free generators xx,...,Xj, and suppose that W cannot be expressed in the form W = Vk for V a word and for k an integer with \k\¥" 1. We ask whether the equation f=W has a solution (xx>... ,Xj) = (a,,... ,Oj) €E GJ whenever G is an infinite symmetric group and/is an element in G. We establish an affirmative answer in… (More)
If /', s are nonzero integers and m is the largest squarefree divisor of rs, 'then for every element ; in the alternating group A " , the equation z = xry' has a solution with x, y e A " , provided that « > 5 and n > (5/2)logm. The bound (5/2)log m improves the bound Am + 1 of Droste. If n > 29, the coefficient 5/2 may be replaced by 2; however, 5/2 cannot… (More)