Beschreibung:
<jats:title>Abstract</jats:title>
<jats:p>In this paper, we investigate the influence of sub-class sizes on a mutually permutable factorized group in which the sub-class sizes of some elements of its factors have certain quantitative properties. Some criteria for a group to be <jats:inline-formula>
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<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0030_eq_001.png" />
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mi>p</m:mi>
</m:math>
<jats:tex-math>p</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>-nilpotent or <jats:inline-formula>
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<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2021-0030_eq_002.png" />
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mi>p</m:mi>
</m:math>
<jats:tex-math>p</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>-supersolvable are given.</jats:p>