Invariable generation of permutation and linear groups

2018/01/30 by Tracey, Gareth M.
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1801.09928

Abstract

A subset \x₁,x₂,\hdots,x_d\ of a group G invariably generates G if \x₁^g₁,x₂^g₂,\hdots,x_d^g_d\ generates G for every d-tuple (g₁,g₂\hdots,g_d)∈ G^d. We prove that a finite completely reducible linear group of dimension n can be invariably generated by \lfloor (3n)/(2)\rfloor elements. We also prove tighter bounds when the field in question has order 2 or 3. Finally, we prove that a transitive [respectively primitive] permutation group of degree n≥ 2 [resp. n≥ 3] can be invariably generated by O(\fracn√logn) [resp. O(\fraclogn√loglogn)] elements.

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• Invariable Generation of Infinite Groups
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