| Summary: | A subgroup <i>H</i> of a finite group <i>G</i> is said to be weakly <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">H</mi> </semantics> </math> </inline-formula>-embedded in <i>G</i> if there exists a normal subgroup <i>T</i> of <i>G</i> such that <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>H</mi> <mi>G</mi> </msup> <mo>=</mo> <mi>H</mi> <mi>T</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>H</mi> <mo>∩</mo> <mi>T</mi> <mo>∈</mo> <mi mathvariant="script">H</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <msup> <mi>H</mi> <mi>G</mi> </msup> </semantics> </math> </inline-formula> is the normal closure of <i>H</i> in <i>G</i>, and <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">H</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is the set of all <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">H</mi> </semantics> </math> </inline-formula>-subgroups of <i>G</i>. In the recent research, Asaad, Ramadan and Heliel gave new characterization of <i>p</i>-nilpotent: <i>Let p be the smallest prime dividing <inline-formula> <math display="inline"> <semantics> <mrow> <mo>|</mo> <mi>G</mi> <mo>|</mo> </mrow> </semantics> </math> </inline-formula>, and P a non-cyclic Sylow p-subgroup of G. Then G is p-nilpotent if and only if there exists a p-power d with <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo><</mo> <mi>d</mi> <mo><</mo> <mo>|</mo> <mi>P</mi> <mo>|</mo> </mrow> </semantics> </math> </inline-formula> such that all subgroups of P of order d and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mi>d</mi> </mrow> </semantics> </math> </inline-formula> are weakly <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">H</mi> </semantics> </math> </inline-formula>-embedded in G</i>. As new applications of weakly <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">H</mi> </semantics> </math> </inline-formula>-embedded subgroups, in this paper, (1) we generalize this result for general prime <i>p</i> and get a new criterion for <i>p</i>-supersolubility; (2) adding the condition “<inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <mi>P</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is <i>p</i>-nilpotent„, here <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <mi>P</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mi>g</mi> <mo>∈</mo> <mi>G</mi> <mo>|</mo> <msup> <mi>P</mi> <mi>g</mi> </msup> <mo>=</mo> <mi>P</mi> <mo>}</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is the normalizer of <i>P</i> in <i>G</i>, we obtain <i>p</i>-nilpotence for general prime <i>p</i>. Moreover, our tool is the weakly <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">H</mi> </semantics> </math> </inline-formula>-embedded subgroup. However, instead of the normality of <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>H</mi> <mi>G</mi> </msup> <mo>=</mo> <mi>H</mi> <mi>T</mi> </mrow> </semantics> </math> </inline-formula>, we just need <inline-formula> <math display="inline"> <semantics> <mrow> <mi>H</mi> <mi>T</mi> </mrow> </semantics> </math> </inline-formula> is <i>S</i>-quasinormal in <i>G</i>, which means that <inline-formula> <math display="inline"> <semantics> <mrow> <mi>H</mi> <mi>T</mi> </mrow> </semantics> </math> </inline-formula> permutes with every Sylow subgroup of <i>G</i>.
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