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• W2056537939 abstract "We study the set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G Superscript left-bracket k right-bracket> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>[</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>G^{[k]}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k Superscript t h> <mml:semantics> <mml:msup> <mml:mi>k</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>t</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>k^{th}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> powers in finite groups <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G Superscript left-bracket 12 right-bracket> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>[</mml:mo> <mml:mn>12</mml:mn> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>G^{[12]}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a subgroup, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must be soluble; moreover, 12 is the minimal number with this property. The proof relies on results of independent interest, classifying almost simple groups <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and positive integers <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G Superscript left-bracket k right-bracket> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>[</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>G^{[k]}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains the socle of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W2056537939 created "2016-06-24" @default.
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• W2056537939 date "2013-09-10" @default.
• W2056537939 modified "2023-09-26" @default.
• W2056537939 title "Powers in finite groups and a criterion for solubility" @default.
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• W2056537939 doi "https://doi.org/10.1090/s0002-9939-2013-11790-9" @default.
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