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• W2433243265 endingPage "982" @default.
• W2433243265 startingPage "959" @default.
• W2433243265 abstract "Abstract The large Conway simple group <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>Co</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> ${{rm Co}_{1}}$ contains a copy of the alternating group <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>A</m:mi> <m:mn>9</m:mn> </m:msub> </m:math> ${{rm A}_{9}}$ and thus contains a nested sequence <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mn>3</m:mn> </m:msub> <m:mo>≤</m:mo> <m:msub> <m:mi>A</m:mi> <m:mn>4</m:mn> </m:msub> <m:mo>≤</m:mo> <m:mi>…</m:mi> <m:mo>≤</m:mo> <m:msub> <m:mi>A</m:mi> <m:mn>9</m:mn> </m:msub> </m:mrow> </m:math> ${{rm A}_{3}leq{rm A}_{4}leqdotsleq{rm A}_{9}}$ . Shortly after <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>Co</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> ${{rm Co}_{1}}$ was discovered, J. G. Thompson recognised that the normalizer of each of the groups in this sequence (apart from that of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>A</m:mi> <m:mn>8</m:mn> </m:msub> </m:math> ${{rm A}_{8}}$ ) is maximal in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>Co</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> ${{rm Co}_{1}}$ and the resulting collection of subgroups <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mn>3</m:mn> <m:mo>⁢</m:mo> <m:mrow /> <m:mo>⁢</m:mo> <m:mpadded> <m:mi>Suz</m:mi> </m:mpadded> </m:mrow> <m:mo>:</m:mo> <m:mn> 2</m:mn> </m:mrow> </m:math> $3mathord{{}^{;textstyle{cdot}}}{rm Suz,{:},2}$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mn>4</m:mn> </m:msub> <m:mo>×</m:mo> <m:msub> <m:mi>G</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>4</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>:</m:mo> <m:mn> 2</m:mn> </m:mrow> </m:math> $({rm A}_{4}times{rm G}_{2}(4)),{:},2$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mn>5</m:mn> </m:msub> <m:mo>×</m:mo> <m:mi>HJ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>:</m:mo> <m:mn> 2</m:mn> </m:mrow> </m:math> $({rm A}_{5}times{rm HJ}),{:},2$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mn>6</m:mn> </m:msub> <m:mo>×</m:mo> <m:msub> <m:mi>U</m:mi> <m:mn>3</m:mn> </m:msub> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>3</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>:</m:mo> <m:mn> 2</m:mn> </m:mrow> </m:math> $({rm A}_{6}times{rm U}_{3}({3})),{:},2$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mn>7</m:mn> </m:msub> <m:mo>×</m:mo> <m:msub> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>7</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>:</m:mo> <m:mn> 2</m:mn> </m:mrow> </m:math> $({rm A}_{7}times{rm L}_{2}({7})),{:},2$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mn>8</m:mn> </m:msub> <m:mo>×</m:mo> <m:msub> <m:mi>S</m:mi> <m:mn>4</m:mn> </m:msub> </m:mrow> </m:math> ${rm A}_{8}times{rm S}_{4}$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mn>9</m:mn> </m:msub> <m:mo>×</m:mo> <m:msub> <m:mi>S</m:mi> <m:mn>3</m:mn> </m:msub> </m:mrow> </m:math> ${rm A}_{9}times{rm S}_{3}$ is now known as the Thompson chain , where Suz denotes the Suzuki simple group and HJ denotes the Hall–Janko group. Remarkably, we can start at the other end in the sense that if we consider <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>U</m:mi> <m:mn>3</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>3</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${{rm U}_{3}({3})}$ in a certain way, we obtain a construction which produces each of the groups <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msub> <m:mi>U</m:mi> <m:mn>3</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>3</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>:</m:mo> <m:mn> 2</m:mn> </m:mrow> </m:math> ${rm U}_{3}({3}),{:},2$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mpadded> <m:mi>HJ</m:mi> </m:mpadded> <m:mo>:</m:mo> <m:mn> 2</m:mn> </m:mrow> </m:math> ${rm HJ},{:},2$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>4</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>:</m:mo> <m:mn> 2</m:mn> </m:mrow> </m:math> ${rm G}_{2}(4),{:},2$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mn>3</m:mn> <m:mo>⁢</m:mo> <m:mrow /> <m:mo>⁢</m:mo> <m:mpadded> <m:mi>Suz</m:mi> </m:mpadded> </m:mrow> <m:mo>:</m:mo> <m:mn> 2</m:mn> </m:mrow> </m:math> $3mathord{{}^{;textstyle{cdot}}}{rm Suz},{:},2$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mn>2</m:mn> <m:mo>×</m:mo> <m:msub> <m:mi>Co</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> </m:math> $2times{rm Co}_{1}$ spontaneously. Indeed, a presentation containing a parameter n is given which, for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>4</m:mn> <m:mo>,</m:mo> <m:mn>5</m:mn> <m:mo>,</m:mo> <m:mn>6</m:mn> <m:mo>,</m:mo> <m:mn>7</m:mn> </m:mrow> </m:mrow> </m:math> ${n=4,5,6,7}$ , defines each of the above groups; n appears just twice in the presentation. Specifically, we associate with each directed edge ij of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>K</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> ${{rm K}_{n}}$ (the complete graph on n vertices) an element <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>t</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> ${t_{ij}}$ of order 7 in some group G , where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>t</m:mi> <m:mrow> <m:mi>j</m:mi> <m:mo>⁢</m:mo> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:msubsup> <m:mi>t</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:math> ${t_{ji}=t_{ij}^{-1}}$ . We insist that G possesses automorphisms corresponding to the symmetric group permuting the n vertices of our <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>K</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> ${{rm K}_{n}}$ , and in addition an automorphism which squares each of the <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>t</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> ${t_{ij}}$ . If we now factor by a relation which ensures that a triangle generates <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>U</m:mi> <m:mn>3</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>3</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${{rm U}_{3}({3})}$ , then a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>K</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> ${{rm K}_{4}}$ generates HJ, a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>K</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> ${{rm K}_{5}}$ generates <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>4</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${{rm G}_{2}(4)}$ , a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>K</m:mi> <m:mn>6</m:mn> </m:msub> </m:math> ${{rm K}_{6}}$ generates <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mn>3</m:mn> <m:mo>⁢</m:mo> <m:mrow /> <m:mo>⁢</m:mo> <m:mi>Suz</m:mi> </m:mrow> </m:math> ${3mathord{{}^{;textstyle{cdot}}}{rm Suz}}$ and a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>K</m:mi> <m:mn>7</m:mn> </m:msub> </m:math> ${{rm K}_{7}}$ generates <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>Co</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> ${{rm Co}_{1}}$ . What happens for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>8</m:mn> </m:mrow> </m:math> ${ngeq 8}$ is explained fully in the text. Thus this is not simply a sequence of nested subgroups in a larger group, but a finite family of closely-related perfect groups." @default.
• W2433243265 created "2016-06-24" @default.
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• W2433243265 date "2016-06-15" @default.
• W2433243265 modified "2023-09-24" @default.
• W2433243265 title "The Thompson chain of subgroups ofbreak the Conway~group Co₁ and complete graphs on ${n}$~vertices" @default.
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