TY - JOUR

T1 - Character bounds for regular semisimple elements and asymptotic results on Thompson’s conjecture

AU - Larsen, Michael

AU - Taylor, Jay

AU - Tiep, Pham Huu

N1 - Publisher Copyright:
© 2023, The Author(s).

PY - 2023/2

Y1 - 2023/2

N2 - For every integer k there exists a bound B= B(k) such that if the characteristic polynomial of g∈ SL n(q) is the product of ≤ k pairwise distinct monic irreducible polynomials over Fq, then every element x of SL n(q) of support at least B is the product of two conjugates of g. We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions (p, q), in the special case that n= p is prime, if g has order qp-1q-1, then every non-scalar element x∈ SL p(q) is the product of two conjugates of g. The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.

AB - For every integer k there exists a bound B= B(k) such that if the characteristic polynomial of g∈ SL n(q) is the product of ≤ k pairwise distinct monic irreducible polynomials over Fq, then every element x of SL n(q) of support at least B is the product of two conjugates of g. We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions (p, q), in the special case that n= p is prime, if g has order qp-1q-1, then every non-scalar element x∈ SL p(q) is the product of two conjugates of g. The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.

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U2 - 10.1007/s00209-022-03193-3

DO - 10.1007/s00209-022-03193-3

M3 - Article

AN - SCOPUS:85146773738

SN - 0025-5874

VL - 303

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 2

M1 - 47

ER -