The Baer–Suzuki width of a complete class of finite group is finite
Abstract
<p> Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a nonempty class of finite groups closed under taking subgroups, homomorphic images, and extensions. According to Gordeev, Grunewald, Kunyavskiĭ, and Plotkin, the Baer–Suzuki width <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper B normal upper S left-parenthesis script upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">B</mml:mi> <mml:mi mathvariant="normal">S</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">X</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {BS}(\mathscr {X})</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="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> does not exceed a nonnegative integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if, in any finite group <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> , the largest normal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -subgroup coincides with the set of elements <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> elements conjugate to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generate an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -subgroup. If there are no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</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="normal upper B normal upper S left-parenthesis script upper X right-parenthesis less-than-or-equal-to m"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">B</mml:mi> <mml:mi mathvariant="normal">S</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">X</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≤ </mml:mo> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathrm {BS}(\mathscr {X})\leq m}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then by definition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper B normal upper S left-parenthesis script upper X right-parenthesis equals normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">B</mml:mi> <mml:mi mathvariant="normal">S</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">X</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathrm {BS}(\mathscr {X})=\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In the paper, it is proved that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper B normal upper S left-parenthesis script upper X right-parenthesis greater-than normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom