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- W4214484297 abstract "Abstract In this paper, we establish sufficient and necessary conditions for the existence of abelian subgroups of maximal order of a finite group <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>G</m:mi> </m:math> G , by means of its commuting graph. The order of these subgroups attains the bound <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>c</m:mi> <m:mo>=</m:mo> <m:mo>∣</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>∣</m:mo> <m:mspace width=-0.25em /> <m:mo>+</m:mo> <m:mo>⋯</m:mo> <m:mo>+</m:mo> <m:mspace width=-0.25em /> <m:mspace width=0.33em /> <m:mo>∣</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>∣</m:mo> </m:math> c=| left[{x}_{1}]| hspace{-0.25em}+cdots +hspace{-0.25em}hspace{0.33em}| left[{x}_{m}]| , where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> left[{x}_{i}] denotes the conjugacy class of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:math> {x}_{i} in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>G</m:mi> </m:math> G and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>m</m:mi> </m:math> m is the smallest integer <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>j</m:mi> </m:math> j such that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∣</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>∣</m:mo> <m:mspace width=-0.25em /> <m:mo>+</m:mo> <m:mo>⋯</m:mo> <m:mo>+</m:mo> <m:mo>∣</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>∣</m:mo> <m:mo>≥</m:mo> <m:mo>∣</m:mo> <m:msub> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mi>G</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∣</m:mo> </m:math> | left[{x}_{1}]| hspace{-0.25em}+cdots +| left[{x}_{j}]| ge | {C}_{G}left({x}_{j})| , where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mi>G</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {C}_{G}left({x}_{j}) is the centralizer of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> {x}_{j} in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>G</m:mi> </m:math> G ." @default.
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- W4214484297 date "2022-01-01" @default.
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- W4214484297 title "Note on structural properties of graphs" @default.
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