Inverse ambiguous functions and automorphisms on finite groups



Abstract

If G is a finite group, then a bijective function f:GG is inverse ambiguous if and only if f(x)-1 = f-1(x) for all xG. We give a precise description when a finite group admits an inverse ambiguous function and when a finite group admits an inverse ambiguous automorphism.


Keywords

inverse function; functional equation; finite groups; abelian groups

1. Y. Berkovich, Groups of Prime Power Order, Vol. 1, De Gruyter Expositions in Mathematics, 46, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
2. R. Cheng, A. Dasgupta, B.R. Ebanks, L.F. Kinch, L.M. Larson and R.B. McFadden, When does f^{-1}=1/f ?, Amer. Math. Monthly 105 (1998), no. 8, 704–717.
3. R. Euler and J. Foran, On functions whose inverse is their reciprocal, Math. Mag. 54 (1981), no. 4, 185–189.
4. M. Griffiths, f(f(x))=x, windmills, and beyond, Math. Mag. 83 (2010), no. 1, 15–23.
5. M. Herzog, Counting group elements of order p modulo p^2, Proc. Amer. Math. Soc. 66 (1977), no. 2, 247–250.
6. H. Kurzweil and B. Stellmacher, The Theory of Finite Groups. An Introduction, Springer-Verlag, New York, 2004.
7. D.J. Schmitz, Inverse ambiguous functions on fields, Aequationes Math. 91 (2017), no. 2, 373–389.
8. D. Schmitz and K. Gallagher, Inverse ambiguous functions on some finite non-abelian groups, Aequationes Math. 92 (2018), no. 5, 963–973.
9. R. Schnabel, Elemente der Gruppentheorie, Mathematik für die Lehrerausbildung, B.G. Teubner, Stuttgart, 1984.
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Published : 2019-06-13


ToborgI. (2019). Inverse ambiguous functions and automorphisms on finite groups. Annales Mathematicae Silesianae, 33, 284-297. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13673

Imke Toborg  imke.toborg@mathematik.uni-halle.de
Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, Germany  Germany



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