The probability of commuting subgroups in arbitrary lattices of subgroups

Muhie, Seid Kassaw; Russo, Francesco G.

doi:10.22108/ijgt.2020.122081.1604

The probability of commuting subgroups in arbitrary lattices of subgroups

Document Type : Research Paper

Authors

Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X1, Rondebosch, 7701, Cape Town, South Africa

10.22108/ijgt.2020.122081.1604

Abstract

A finite group $G$‎, ‎in which two randomly chosen subgroups $H$ and $K$ commute‎, ‎has been classified by Iwasawa in 1941‎. ‎It is possible to define a probabilistic notion‎, ‎which ``measures the distance'' of $G$ from the groups of Iwasawa‎. ‎Here we introduce the generalized subgroup commutativity degree $gsd(G)$ for two arbitrary sublattices $\mathrm{S}(G)$ and $\mathrm{T}(G)$ of the lattice of subgroups $\mathrm{L}(G)$ of $G$‎. ‎Upper and lower bounds for $gsd(G)$ are shown and we study the behaviour of $gsd(G)$ with respect to subgroups and quotients‎, ‎showing new numerical restrictions‎.

Keywords

20.1001.1.22517650.2021.10.3.3.6

Main Subjects

20P05 Probabilistic methods in group theory

References

[1] S. Eberhard, Commuting probabilities of finite groups, Bull. London Math. Soc., 47 (2015) 769–808.

[2] P. Erd˝os and P. Tur´an, On some problems of a statistical group theory I, Z. Wahrscheinlichkeitstheorie und Verw.
Gebiete, 4 (1965) 175–186.

[3] A. Erfanian, P. Lescot and R. Rezaei, On the relative commutativity degree of a subgroup of a finite group, Comm.
Algebra, 35 (2007) 4183–4197.

[4] S. Garion and A. Shalev, Commutator maps, measure preservation, and T-systems, Trans. Amer. Math. Soc., 361
(2009) 4631–4651.

[5] W. Herfort, K. H. Hofmann and F. G. Russo, Periodic Locally Compact Groups, de Gruyter, Berlin, 2018.

[6] K. H. Hofmann and F. G. Russo, The probability that x and y commute in a compact group, Math. Proc. Camb.
Philos. Soc., 153 (2012) 557–571.

[7] K. Iwasawa, Uber die endlichen Gruppen und die Verb¨ande ihrer Untergruppen, ¨ J. Fac. Sci. Imp. Univ. Tokyo,
Sect. I, 4 (1941) 171–199.

[8] E. Kazeem, Subgroup commutativity degree of profinite groups, Topology Appl., 263 (2019) 1–15.

[9] M. S. Lazorec, Probabilistic aspects of ZM-groups, Comm. Algebra, 47 (2018) 541–552.

[10] M. S. Lazorec, Relative cyclic subgroup commutativity degrees of finite groups, Filomat, 33 (2019) 4021–4032.

[11] D. E. Otera and F. G. Russo, Subgroup S-commutativity degrees of finite groups, Bull. Belg. Math. Soc. Simon
Stevin, 19 (2012) 373–382.

[12] F. G. Russo, Strong subgroup commutativity degree and some recent problems on the commuting probabilities of
elements and subgroups, Quaest. Math., 39 (2016) 1019–1036.

[13] R. Schmidt, Subgroup Lattices of Groups, de Gruyter, Berlin, 1994.

[14] A. Shalev, Some Results and Problems in the Theory of Word Maps, Erd˝os Centennial, L. Lov´asz , I. Z. Ruzsa,
V. T. S´os and (eds), Bolyai Society Mathematical Studies, 25, Springer, Berlin, 2013 611–649.

[15] M. Tˇarnˇauceanu, Subgroup commutativity degrees of finite groups, J. Algebra, 321 (2009) 2508–2520.

[16] M. Tˇarnˇauceanu, Addendum to: Subgroup commutativity degrees of finite groups, J. Algebra, 337 (2011) 363–368.

[17] M. Tˇarnˇauceanu, An arithmetic method of counting the subgroups of a finite abelian group, Bull. Math. Soc. Sci.
Math. Roumanie, 53 (2010) 373–386.