| Summary: | Given a positive integer <i>n</i>, a finite group <i>G</i> is called quasi-core-<i>n</i> if <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>〈</mo> <mi>x</mi> <mo>〉</mo> </mrow> <mo>/</mo> <msub> <mrow> <mo>〈</mo> <mi>x</mi> <mo>〉</mo> </mrow> <mi>G</mi> </msub> </mrow> </semantics> </math> </inline-formula> has order at most <i>n</i> for any element <i>x</i> in <i>G</i>, where <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo>〈</mo> <mi>x</mi> <mo>〉</mo> </mrow> <mi>G</mi> </msub> </semantics> </math> </inline-formula> is the normal core of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>〈</mo> <mi>x</mi> <mo>〉</mo> </mrow> </semantics> </math> </inline-formula> in <i>G</i>. In this paper, we investigate the structure of finite quasi-core-<i>p p</i>-groups. We prove that if the nilpotency class of a quasi-core-<i>p p</i>-group is <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>+</mo> <mi>m</mi> </mrow> </semantics> </math> </inline-formula>, then the exponent of its commutator subgroup cannot exceed <inline-formula> <math display="inline"> <semantics> <msup> <mi>p</mi> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </semantics> </math> </inline-formula>, where <i>p</i> is an odd prime and <i>m</i> is non-negative. If <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula>, we prove that every quasi-core-3 3-group has nilpotency class at most 5 and its commutator subgroup is of exponent at most 9. We also show that the Frattini subgroup of a quasi-core-2 2-group is abelian.
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