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Proceedings of the American Mathematical Society
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
MathSciNet review: 813802
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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$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0813802-7
PII: S 0002-9939(1986)0813802-7
Article copyright: © Copyright 1986 American Mathematical Society