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• W3035462393 abstract "Abstract Let <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>cs</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {mathrm{cs}(G)} denote the set of conjugacy class sizes of a group G , and let <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msup> <m:mi>cs</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi>cs</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>∖</m:mo> <m:mrow> <m:mo stretchy=false>{</m:mo> <m:mn>1</m:mn> <m:mo stretchy=false>}</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> mathrm{cs}^{*}(G)=mathrm{cs}(G)setminus{1} be the sizes of non-central classes. We prove three results. We classify all finite groups for which (1) <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>cs</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy=false>{</m:mo> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>+</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mi mathvariant=normal>…</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>+</m:mo> <m:mrow> <m:mi>r</m:mi> <m:mo>⁢</m:mo> <m:mi>d</m:mi> </m:mrow> </m:mrow> <m:mo stretchy=false>}</m:mo> </m:mrow> </m:mrow> </m:math> {mathrm{cs}(G)={a,a+d,dots,a+rd}} is an arithmetic progression with <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>r</m:mi> <m:mo>⩾</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {rgeqslant 2} ; (2) <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msup> <m:mi>cs</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy=false>{</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>4</m:mn> <m:mo>,</m:mo> <m:mn>6</m:mn> <m:mo stretchy=false>}</m:mo> </m:mrow> </m:mrow> </m:math> {mathrm{cs}^{*}(G)={2,4,6}} is the smallest case where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>cs</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {mathrm{cs}^{*}(G)} is an arithmetic progression of length more than 2 (our most substantial result); (3) the largest two members of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>cs</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {mathrm{cs}^{*}(G)} are coprime. For (3), it is not obvious, but it is true that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>cs</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {mathrm{cs}^{*}(G)} has two elements, and so is an arithmetic progression." @default.
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• W3035462393 title "Conjugacy class sizes in arithmetic progression" @default.
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