A new kind of polynomials for finite groups

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2024年12月5日

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A. K. Asboei,
C. S. Anabanti

摘要

Let Property X be a certain property of some finite groups; for instance, nilpotent, supersolvable, solvable et cetera. The Thompson-like problem asks whether for two finite groups $G_{1}$ $G_{1}$ and $G_{2}$ $G_{2}$ of the same order type, does $G_{2}$ $G_{2}$ always satisfy Property X if $G_{1}$ $G_{1}$ satisfies Property X? Thompson-like problem has been solved where Property X is nilpotent. In this paper, we will introduce a new kind of polynomials for finite groups, which we shall call ‘the order polynomials’ and use it to propose another way of solving the Thompson-like problem, where Property X is nilpotent. Furthermore, we examine Thompson-like problem when Property X is supersolvable, and give infinitely many counterexamples to show that the answer to the problem is in the negative there.