How abelian can a non-abelian group be?

  • Katarzyna Słomczyńska Institute of Mathematics, Pedagogical University of Cracow, Poland

Abstract

In this paper we survey, also in historical perspective, the results connected with the notion of the commutativity degree of a finite group, i.e., the probability that two randomly selected elements of the group commute.

References

Antolín, Y., Martino, A., Ventura, E.: 2017, Degree of commutativity of infinite groups, Proc. Amer. Math. Soc. 145, 479–485.

Barry, F., MacHale, D., Ní Shé, Á.: 2006, Some supersolvability conditions for finite groups, Math. Proc. R. Ir. Acad. 106A, 163–177.

Baumeister, B., Maróti, A., Tong-Viet, H. P.: 2016, Finite groups have more conjugacy classes, Forum Math. 29, 259–275.

Bertram, E. A.: 2013, New reductions and logarithmic lower bounds for the number of conjugacy classes in finite groups, Bull. Austral. Math. Soc. 87, 406–424.

Buckley, S. M., MacHale, D.: 2017, Contrasting the commuting probabilities of groups and rings, preprint, http://archive.maths.nuim.ie/staff/sbuckley/Papers/bm_g-vs-r.pdf.

Castelaz, A.: 2010, Commutativity degree of finite groups, Master’s thesis, Wake Forest University.

Das, A. K., Nath, R. K., Pournaki, M. R.: 2013, A survey on the estimation of commutativity in finite groups, Southeast Asian Bull. Math. 37, 161–180.

Dixon, J. D.: 2002, Probabilistic group theory, C. R. Math. Acad. Sci. Soc. R. Can. 24, 1–15.

Eberhard, S.: 2015, Commuting probabilities of finite groups, Bull. London Math. Soc. 47, 796–808.

Erdös, P., Turán, P.: 1968, On some problems of a statistical group-theory, iv, Acta Math. Acad. Sci. Hung. 19, 413–435.

Gallagher, P. X.: 1970, The number of conjugacy classes in a finite group, Math. Z. 118, 175–179.

Gallian, J. A.: 2013, Contemporary Abstract Algebra, 8th ed., Belmont, CA, Cengage Learning.

Guralnick, R. M., Robinson, G. R.: 2006, On the commuting probability in finite groups, J. Algebra 300, 509–528.

Gustafson, W. H.: 1973, What is the probability that two group elements commute?, Amer. Math. Monthly 80, 1031–1034.

Hegarty, P.: 2013, Limit points in the range of the commuting probability function on finite groups, J. Group Theory 16, 235–247.

Joseph, K. S.: 1969, Commutativity in non-abelian groups, PhD thesis, UCLA.

Joseph, K. S.: 1977, Several conjectures on commutativity in algebraic structures, Amer. Math. Monthly 84, 550–551.

Landau, E.: 1903, Über die Klassenzahl binären quadratischen Formen von negativer Discriminante, Math. Ann. 56, 671–676.

Leavitt, J. L., Sherman, G. J., Walker, M. E.: 1992, Rewriteability in finite groups, Amer. Math. Monthly 99, 446–452.

Lescot, P.: 1978, Sur certains groupes finis, Rev. Math. Spéciales, Avril 1987, 276–277.

Lescot, P.: 1988, Degré de commutativité et structure d’un groupe fini (1), Rev. Math. Spéciales, Avril 1988, 276–279.

MacHale, D.: 1974, How commutative can a non-commutative group be?, Math. Gazette 58, 199–202.

MacHale, D.: 1976, Commutativity in finite rings, Amer. Math. Monthly 83, 30–32.

MacHale, D.: 1990, Probability in finite semigroups, Irish Math. Soc. Bull. 25, 64–68.

Miller, G. A.: 1919, Groups possessing a small number of sets of conjugate operators, Trans. Amer. Math. Soc. 20, 260–270.

Published
2018-06-01
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