V.M. Leontiev. On the exponents of commutators from P. Hall’s collection formula ... P. 182-198

Let $G$ be a group, and let $x,y \in G$. We find an explicit form of the exponents of some commutators from P. Hall's collection formula for the expression $(xy)^n$, $n \in \mathbb{N}$. The exponents for the series of commutators $[y,\!\!\ _ux,\!\!\ _vy]$ and $[[y,\!\!\ _ux],[y,\!\!\ _vx]]$ are found in the Hall form, i.e., in the form of integer-valued polynomials in $n$ with zero constant term, and also modulo $n$ when $n$ is a prime number. The exponents for the series of commutators $[[y,\!\!\ _{u}x,\!\!\ _{v}y],\!\!\ _{t_{1}}[y,\!\!\ _{u_1}x,\!\!\ _{v_1}y], \ldots,\!\!\ _{t_{h}}[y,\!\!\ _{u_h}x,\!\!\ _{v_h}y]]$ are found in the form of multiple combinatorial sums. As a consequence, we obtain an explicit form of Hall's collection formula in two cases: the group $G$ has solvability length 2, the commutator subgroup $G'$ has nilpotency class 2, and $y \in C_G(G')$. A collection formula for the expression $(xy)^n$ is obtained in an explicit form when the group $G$ has solvability length 3. To obtain these results we parameterize the uncollected part of the collection formula by the binary weight function. The results may be useful in solving problems in combinatorial group theory and in studying the regularity of finite $p$-groups.

Keywords: collection process, collection formula, commutator

Received September 16, 2021

Revised November 22, 2021

Accepted November 29, 2021

Funding Agency:  This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075-02-2022-876).

Vladimir Markovich Leontiev, doctoral student, Institute of Mathematics and Computer Science of the Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: v.m.leontiev@outlook.com

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Cite this article as: V.M. Leontiev. On the exponents of commutators from P. Hall’s collection formula, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 1, pp. 182–198.