Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The universality of words $ x\sp ry\sp s$ in alternating groups


Authors: J. L. Brenner, R. J. Evans and D. M. Silberger
Journal: Proc. Amer. Math. Soc. 96 (1986), 23-28
MSC: Primary 20F10; Secondary 20B35
DOI: https://doi.org/10.1090/S0002-9939-1986-0813802-7
MathSciNet review: 813802
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If $ r,s$ are nonzero integers and $ m$ is the largest squarefree divisor of $ rs$, then for every element $ z$ in the alternating group $ {A_n}$, the equation $ z = {x^r}{y^s}$ has a solution with $ x,y \in {A_n}$, provided that $ n \geqslant 5$ and $ n \geqslant (5/2)\log m$. The bound $ (5/2)\log m$ improves the bound $ 4m + 1$ of Droste. If $ n \geqslant 29$, the coefficient $ 5/2$ may be replaced by 2; however, $ 5/2$ cannot be replaced by 1 even for all large $ n$.


References [Enhancements On Off] (What's this?)

  • [1] E. Bertram, Even permutations as a product of two conjugate cycles, J. Combin. Theory (A) 12 (1972), 368-380. MR 0297853 (45:6905)
  • [2] G. Boccara, Décompositions d'une permutation d'un ensemble fini en produit de deux cycles, Discrete Math. 23 (1978), 189-205. MR 523070 (81c:05010)
  • [3] J. L. Brenner, Covering theorems for finite nonahelian simple groups. IX, How the square of a class with two nontrivial orbits in $ {S_n}$ covers $ {A_n}$, Ars Combin. 4 (1977), 151-176. MR 0576549 (58:28162a)
  • [4] J. L. Brenner and R. J. Evans, Even permutations as a product of two elements of order 5, Preprint 1985.
  • [5] J. L. Brenner and I. Riddell, Noncanonical factorization of a permutation, Amer. Math. Monthly 84 (1977), 39-40. MR 1538247
  • [6] M. Droste, On the universality of words for the alternating groups. Proc. Amer. Math. Soc. 96 (1986), 18-22. MR 813801 (87c:20063)
  • [7] A. Ehrenfeucht, S. Fajtlowicz, J. Malitz, and J. Mycielski, Some problems on the universality of words in groups, Algebra Universalis 11 (1980), 261-263. MR 588219 (81k:20045)
  • [8] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94. MR 0137689 (25:1139)
  • [9] D. M. Silberger, For $ k$ big the word $ {x^m}{y^n}$ is universal for $ {A_k}$, Abstracts Amer. Math. Soc. 3 (1982), 293.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20F10, 20B35

Retrieve articles in all journals with MSC: 20F10, 20B35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0813802-7
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society