2009/12/02 by Paul Flavell, Simon Guest, Simon D. Guest +2 · 3 citations
Mathematics · Neuroscience · #Finite Group Theory Research #Nuclear Receptors and Signaling
paper · pdf · doi:10.1090/s0002-9939-09-10066-7
We prove that there exists a constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the property: if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a conjugacy class of a finite group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> elements of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generate a solvable subgroup, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals 4"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k=4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also present proofs that do not use the Classification Theorem. The most direct proof gives a value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals 10"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>10</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k=10</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . By lengthening one of our arguments slightly, we obtain a value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals 7"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k=7</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .