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- W3094993459 abstract "Abstract A subgroup H of a group G is said to be conditionally permutable (or c -permutable for short) in G if, for every subgroup T of G , there exists an element <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> {xin G} such that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>H</m:mi> <m:mo></m:mo> <m:msup> <m:mi>T</m:mi> <m:mi>x</m:mi> </m:msup> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>T</m:mi> <m:mi>x</m:mi> </m:msup> <m:mo></m:mo> <m:mi>H</m:mi> </m:mrow> </m:mrow> </m:math> {HT^{x}=T^{x}H} . A subgroup H of a group G is said to be completely c -permutable in G if, for every subgroup T of G , the subgroups H and T are c -permutable in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo stretchy=false>〈</m:mo> <m:mi>H</m:mi> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy=false>〉</m:mo> </m:mrow> </m:math> {langle H,Trangle} . In this paper, we prove that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>H</m:mi> <m:mo>/</m:mo> <m:msub> <m:mi>H</m:mi> <m:mi>G</m:mi> </m:msub> </m:mrow> </m:math> {H/H_{G}} is nilpotent if H is a completely c -permutable subnormal subgroup of G . This result generalizes a well-known theorem of Ito and Szép, and gives a positive answer to an open problem in [W. Guo, Structure Theory for Canonical Classes of Finite Groups, Springer, Heidelberg, 2015]. We also use complete c -permutability to determine the p -supersolubility of a group G ." @default.
- W3094993459 created "2020-11-09" @default.
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- W3094993459 date "2020-10-24" @default.
- W3094993459 modified "2023-09-28" @default.
- W3094993459 title "On a theorem of Ito and Szép" @default.
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- W3094993459 doi "https://doi.org/10.1515/jgth-2020-0084" @default.
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