Type: Article
Publication Date: 2019-04-06
Citations: 0
DOI: https://doi.org/10.1515/jgth-2018-0188
Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Irr</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> {\operatorname{Irr}(G)} denote the set of complex irreducible characters of a finite group G , and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>cd</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> {\operatorname{cd}(G)} be the set of degrees of the members of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Irr</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> {\operatorname{Irr}(G)} . For positive integers k and l , we say that the finite group G has the property <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="script">𝒫</m:mi> <m:mi>k</m:mi> <m:mi>l</m:mi> </m:msubsup> </m:math> {\mathcal{P}^{l}_{k}} if, for any distinct degrees <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>a</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>a</m:mi> <m:mi>k</m:mi> </m:msub> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:mi>cd</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:mrow> </m:math> {a_{1},a_{2},\dots,a_{k}\in\operatorname{cd}(G)} , the total number of (not necessarily different) prime divisors of the greatest common divisor <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>gcd</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>a</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>a</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>a</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\gcd(a_{1},a_{2},\dots,a_{k})} is at most <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>l</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {l-1} . In this paper, we classify all finite almost simple groups satisfying the property <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="script">𝒫</m:mi> <m:mn>3</m:mn> <m:mn>2</m:mn> </m:msubsup> </m:math> {\mathcal{P}_{3}^{2}} . As a consequence of our classification, we show that if G is an almost simple group satisfying <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="script">𝒫</m:mi> <m:mn>3</m:mn> <m:mn>2</m:mn> </m:msubsup> </m:math> {\mathcal{P}_{3}^{2}} , then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:mi>cd</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 fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mo>⩽</m:mo> <m:mn>8</m:mn> </m:mrow> </m:math> {\lvert\operatorname{cd}(G)\rvert\leqslant 8} .
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