Abstract
The aim of this paper is to give a generalization of the concept of commutativity degree of a finite group G (denoted by d(G)), to the concept of commutativity degree of a group algebra F[G], where G is a finite group and F is a finite field. We prove that two isoclinic groups for which the order of their centers are equal have the same commutativity degree. Finally, we give some lower and upper bounds for the commutativity degree of group algebra F[G] in terms of the order of G, the order of F and d(G).
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Erdös, P., Turán, P.: On some problems of a statistical group-theory IV. Acta Math. Acad. Sci. Hung. 19, 413–435 (1968)
Erfanian, A., Rezaei, R., Lescot, P.: On the relative commutativity degree of a subgroup of a finite group. Commun. Algebra 35, 4183–4197 (2007)
Erfanian, A., Rezaei, R., Russo, F.G.: Relative \(n\)-isoclinism classes and relative \(n\)-th nilpotency degree of finite groups. Filomat 27(2), 367–371 (2013)
Gallagher, P.X.: The number of conjugacy classes in a finite group. Math. Z. 118, 175–179 (1970)
Gustafson, W.H.: What is the probability that two groups elements commute? Am. Math. Mon. 80, 1031–1034 (1973)
Hall, P.: The classification of prime-power groups. J. Reine Angew. Math. 182, 130–141 (1940)
Lescot, P.: Isoclinism classes and commutativity degrees of finite groups. J. Algebra 177(3), 847–869 (1995)
Lescot, P.: Central extensions and commutativity degree. Commun. Algebra 29, 4451–4460 (2001)
Miller, G.A.: The second homology group of a group. Proc. Am. Math. Soc. 3, 588–595 (1952)
Passman, D.S.: The Algebraic Structure of Group Rings. Krieger Publishing, Malabar (1985)
Rezaei, R., Erfanian, A.: On the commutativity degree of compact groups. Arch. Math. (Basel) 93(4), 201–212 (2009)
Robinson, D.J.S.: A Course in the Theory of Groups, 2nd edn. Springer, New York (1995)
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The authors would like to thank anonymous referees for providing us helpful and constructive comments and suggestions.
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Chashiani, A., Rezaei, R. On the commutativity degree of a group algebra. Afr. Mat. 32, 1137–1145 (2021). https://doi.org/10.1007/s13370-021-00887-5
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DOI: https://doi.org/10.1007/s13370-021-00887-5