TY - JOUR
T1 - Surjective word maps and Burnside’s paqb theorem
AU - Guralnick, Robert M.
AU - Liebeck, Martin W.
AU - O’Brien, E. A.
AU - Shalev, Aner
AU - Tiep, Pham Huu
N1 - Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2018/8/1
Y1 - 2018/8/1
N2 - We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map (x, y) ↦ xNyN is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map (x, y, z) ↦ xNyNzN is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit–Thompson. We also prove asymptotic results about the surjectivity of the word map (x, y) ↦ xNyN that depend on the number of prime factors of the integer N.
AB - We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map (x, y) ↦ xNyN is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map (x, y, z) ↦ xNyNzN is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit–Thompson. We also prove asymptotic results about the surjectivity of the word map (x, y) ↦ xNyN that depend on the number of prime factors of the integer N.
UR - http://www.scopus.com/inward/record.url?scp=85044972585&partnerID=8YFLogxK
U2 - 10.1007/s00222-018-0795-z
DO - 10.1007/s00222-018-0795-z
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AN - SCOPUS:85044972585
SN - 0020-9910
VL - 213
SP - 589
EP - 695
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 2
ER -