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- W2468160556 abstract "Abstract A finite group G is said to be n -equilibrated if, for all subgroups H and K of G with <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>d</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>H</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>≥</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> ${d(H)geq n}$ and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>d</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>K</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>≥</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> ${d(K)geq n}$ , either <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>H</m:mi> <m:mo>≤</m:mo> <m:mrow> <m:msub> <m:mi>N</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>K</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> ${Hleq N_{G}(K)}$ or <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>K</m:mi> <m:mo>≤</m:mo> <m:mrow> <m:msub> <m:mi>N</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>H</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> ${Kleq N_{G}(H)}$ . In this paper, we investigate the structure of finite 2-equilibrated p -groups G . We provide a complete classification of such groups when G is 2-generator. If G is 3-generator, we prove that G has a normal metacyclic subgroup N such that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> ${G/N}$ is cyclic and, if G has at least four generators, we prove that G is modular and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>G</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>≤</m:mo> <m:mrow> <m:mo>〈</m:mo> <m:mi>x</m:mi> <m:mo>〉</m:mo> </m:mrow> </m:mrow> </m:math> ${G^{prime}leqlangle xrangle}$ for any element x in G with <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>o</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>exp</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> ${o(x)=exp(G)}$ ." @default.
- W2468160556 created "2016-07-22" @default.
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- W2468160556 date "2016-01-08" @default.
- W2468160556 modified "2023-09-24" @default.
- W2468160556 title "On finite 2-equilibrated p-groups" @default.
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- W2468160556 doi "https://doi.org/10.1515/jgth-2015-0042" @default.
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