Тип публикации: статья из журнала
Год издания: 2022
Идентификатор DOI: 10.21538/0134-4889-2022-28-1-182-198
Ключевые слова: collection formula, collection process, commutator
Аннотация: Let G be a group, and let x, y ∈ G. We find an explicit form of the exponents of some commutators from P. Hall’s collection formula for the expression (xy)n, n ∈ 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, ux, vy], t1[y, u1x, v1y], . . ., th[y, uhx, vhy]] 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 ∈ CG(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. © 2022 Krasovskii Institute of Mathematics and Mechanics. All Rights Reserved.
Журнал: Trudy Instituta Matematiki i Mekhaniki UrO RAN
Выпуск журнала: Vol. 28, Is. 1
Номера страниц: 182-198
ISSN журнала: 01344889
Издатель: Krasovskii Institute of Mathematics and Mechanics