Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter March 19, 2020

Two collection formulas

  • Sergey Kolesnikov , Vladimir Leontiev and Georgy Egorychev
From the journal Journal of Group Theory

Abstract

Let G be a group, x , y G , and H is a normal closure in G of the subgroup generated by all commutators of x , y that include more than two occurrences of y. Our main result is a parametrization of the uncollected part of Hall’s collection formula for ( x y ) n and two different collection formulas modulo H in an explicit form. In particular, we found explicit expressions for the exponents of the commutators in Hall’s collection formula for both known cases and some unknown cases. Also we give combinatorial identities, which simplify those expressions and help to investigate their divisibility.


Communicated by Nigel Boston


References

[1] G. P. Egorychev, Integral Representation and the Computation of Combinatorial Sums, Transl. Math. Monogr. 59, American Mathematical Society, Providence, 1984. 10.1090/mmono/059Search in Google Scholar

[2] G. P. Egorychev, Method of coefficients: an algebraic characterization and recent applications, Advances in Combinatorial Mathematics, Springer, Berlin (2009), 1–30. 10.1007/978-3-642-03562-3_1Search in Google Scholar

[3] G. P. Egorychev, S. G. Kolesnikov and V. M. Leontiev, Two collection formulas, International Algebraic Conference dedicated to the 110th anniversary of Professor A. G. Kurosh (1908–1971), 2018, https://lomonosov-msu.ru/file/event/4623/eid4623\_attach\_e9e90e270f009df12cdd7d3771ec9d3395156bb4.pdf. Search in Google Scholar

[4] M. Hall, Jr., The Theory of Groups, The Macmillan, New York, 1959. 10.4159/harvard.9780674592711Search in Google Scholar

[5] P. Hall, A contribution to the theory of groups of prime-power order, Proc. Lond. Math. Soc. (2) 36 (1934), 29–95. 10.1112/plms/s2-36.1.29Search in Google Scholar

[6] E. F. Krause, On the collection process, Proc. Amer. Math. Soc. 15 (1964), 497–504. 10.1090/S0002-9939-1964-0165008-0Search in Google Scholar

[7] V. M. Leontiev, P. Hall’s collection formulas with some restrictions on commutator subgroup (in Russian), August Möbius Contest, 2016, http://www.moebiuscontest.ru/files/2016/leontiev.pdf. Search in Google Scholar

[8] A. I. Skopin, The collecting formula (in Russian), Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 46 (1974), 59–63. Search in Google Scholar

[9] A. I. Skopin, The Jacobi identity and P. Hall’s collection formula for transmetabelian groups of two types, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 175 (1989), 106–112. 10.1007/BF01100121Search in Google Scholar

Received: 2019-05-23
Revised: 2020-01-30
Published Online: 2020-03-19
Published in Print: 2020-07-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 3.6.2023 from https://www.degruyter.com/document/doi/10.1515/jgth-2019-0074/html
Scroll to top button