Autocommutativity degree for topological groups

Document Type : Original Article

Author
Seyyed Ali Moosavi
Department of Mathematics, Faculty of Basic science, University of Qom, Qom, Iran
10.22034/maco.5.1.3
Abstract
Let G be a compact Hausdorff topological group and suppose that Aut(G) be
the group of topological automorphisms of G, which itself is a compact Hausdorff topological
group. In this paper, we will define the notion of autocommutativity degree for the group
G, which generalizes the concept of autocommutativity degree, in the case of finite groups.
We will prove some properties of the autocommutativity degree for topological groups. In
particular, we will state an upper bound for the autocommutativity degree of non-abelian
groups and investigate the structure of groups that attain this upper bound. Also we provide some examples of inifinite topological groups and their autocommutativity degree.

Keywords

Subjects

[1] H. Arora and R. Karan, What is the probability an automorphism fixes a group element, Comm. Algebra, 45:1141–1150, 2017.
[2] A. Baker, Matrix Groups: An Introduction to Lie Group Theory, Springer Undergraduate Mathematics Series, Springer Verlag, London, 2002.
[3] A. Erfanian, F.G. Russo, Probability of mutually commuting n-tuples in some classes of compact groups, Bulletin of the Iranian Mathematical Society, 34(2):27–37, 2008.
[4] S.A. Moosavi, Relative commutativity degree for some topological groups, Measure Algebras and Appli-cations, 1(1):1–-12, 2023.
[5] L. Nachbin, The Haar Integral, Van Nostrand, Princeton, 1965.
[6] R. Rezaei, F.G. Russo, Bounds for the relative n-th nilpotency degree in compact groups, Asian-European Journal of Mathematics, 4:495–506, 2011.
[7] D. Robinson, A Course in the Theory of Groups, Germany:Springer, New York, 1966.
[8] W.H. Gustafson, What is the probability that two groups elements commute?, Amer. Math. Monthly, 80: 1031–1304, 1973.
[9] M.R. Rismanchian and Z. Sepehrizadeh, Autoisoclinism classes and autocommutativity degree of finite groups, Hacet. J. Math. Stat., 44:893–899, 2015.
[10] G. J. Sherman, What is the probability an automorphism fixes a group element?, Amer. Math. Monthly, 82:261–264, 1975.