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• W2011052296 abstract "We give a new, concise definition of the Conway group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=dot normal upper O> <mml:semantics> <mml:mrow> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>O</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>cdot mathrm {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as follows. The Mathieu group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper M 24> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>M</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>24</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>mathrm {M}_{24}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartBinomialOrMatrix 24 Choose 4 EndBinomialOrMatrix> <mml:semantics> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac linethickness=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>24</mml:mn> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>4</mml:mn> </mml:mrow> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>left ({24}atop {4} right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> tetrads. We use this action to define a progenitor <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding=application/x-tex>P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of shape <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 Superscript star StartBinomialOrMatrix 24 Choose 4 EndBinomialOrMatrix Baseline colon normal upper M 24> <mml:semantics> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>⋆<!-- ⋆ --></mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac linethickness=0> <mml:mn>24</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:msup> <mml:mo>:</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>M</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>24</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>2^{star left ( 24 atop 4 right )}:mathrm {M}_{24}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; that is, a free product of cyclic groups of order 2 extended by a group of permutations of the involutory generators. A simple lemma leads us directly to an easily described, short relator, and factoring <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding=application/x-tex>P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by this relator results in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=dot normal upper O> <mml:semantics> <mml:mrow> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>O</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>cdot mathrm {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Consideration of the lowest dimension in which <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=dot normal upper O> <mml:semantics> <mml:mrow> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>O</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>cdot mathrm {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can act faithfully produces Conway’s elements <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi Subscript upper T> <mml:semantics> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>T</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>xi _T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the 24–dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=dot normal upper O> <mml:semantics> <mml:mrow> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>O</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>cdot mathrm {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the integral vectors in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R 24> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>24</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>{mathbb R}_{24}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2011052296 title "The Leech lattice Λ and the Conway group ⋅𝑂 revisited" @default.
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