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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2026. Vol 57

The Collection Formula and Regular 𝑝-Groups

Author(s)

Vladimir M. Leontiev, Yaroslav S. Fiskov

Siberian Federal University, Krasnoyarsk, Russian Federation

Abstract
For any elements 𝑥, 𝑦 of a group 𝐺 and for any positive integer 𝑛, we obtain the collection formula for the 𝑛-th power of the product 𝑥𝑦 when all commutators in 𝐺 that have more than 3 occurrences of the element 𝑦 are equal to 1. Suppose 𝑛 = 𝑝 is a prime number, and any commutator weighting more than 𝑝 − 1 has an order 1 or 𝑝. In terms of the segment of the formula including only commutators of weight 𝑝 with the exponents modulo 𝑝, we provide a necessary condition for the regularity of a finite 𝑝-group. This result generalizes known necessary conditions that were used in proving the non-regularity of some matrix 𝑝-groups.
About the Authors

Vladimir M. Leontiev, Cand. Sci. (Phys.–Math.), Assoc. Prof., Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, v.m.leontiev@outlook.com

Yaroslav S. Fiskov, Student, Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, yarfis04@gmail.com

For citation
V. M. Leontiev, Ya. S. Fiskov The Collection Formula and Regular 𝑝- Groups. The Bulletin of Irkutsk State University. Series Mathematics, 2026, vol. 57, pp. 113–127. https://doi.org/10.26516/1997-7670.2026.57.113
Keywords
P. Hall’s collection formula, collection process, commutator, regular 𝑝-group
UDC
512.54
MSC
20F12
DOI
https://doi.org/10.26516/1997-7670.2026.57.113
References
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