TY - JOUR

T1 - Effective results on the waring problem for finite simple groups

AU - Guralnick, Robert M.

AU - Tiep, Pham Huu

N1 - Publisher Copyright:
© 2015 by Johns Hopkins University Press.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2015/10

Y1 - 2015/10

N2 - Let G be a finite quasisimple group of Lie type. We show that there are regular semi simple elements x, y ∈ G, x of prime order, and |y| is divisible by at most two primes, such that (formula presented). In fact in all but four cases, y can be chosen to be of square-free order. Using this result, we prove an effective version of a previous result of Larsen, Shalev, and Tiep by showing that, given any integer m ≥ 1, if the order of a finite simple group S is at least (formula presented), then every element in S is a product of two mth powers. Furthermore, the verbal width of xm on any finite simple group S is at most (formula presented). We also show that, given any two non-trivial words w1, w2, if G is a finite quasi simple group of large enough order, then (formula presented).

AB - Let G be a finite quasisimple group of Lie type. We show that there are regular semi simple elements x, y ∈ G, x of prime order, and |y| is divisible by at most two primes, such that (formula presented). In fact in all but four cases, y can be chosen to be of square-free order. Using this result, we prove an effective version of a previous result of Larsen, Shalev, and Tiep by showing that, given any integer m ≥ 1, if the order of a finite simple group S is at least (formula presented), then every element in S is a product of two mth powers. Furthermore, the verbal width of xm on any finite simple group S is at most (formula presented). We also show that, given any two non-trivial words w1, w2, if G is a finite quasi simple group of large enough order, then (formula presented).

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U2 - 10.1353/ajm.2015.0035

DO - 10.1353/ajm.2015.0035

M3 - Article

AN - SCOPUS:84942860461

VL - 137

SP - 1401

EP - 1430

JO - American Journal of Mathematics

JF - American Journal of Mathematics

SN - 0002-9327

IS - 5

ER -