Aequationes mathematicae

, Volume 92, Issue 5, pp 963–973 | Cite as

Inverse ambiguous functions on some finite non-abelian groups

  • David SchmitzEmail author
  • Katherine Gallagher


In Schmitz (Aequ Math 91:373–389, 2017), the first author defines an “inverse ambiguous function” on a group G to be a bijective function \(f:G \rightarrow G\) satisfying the functional equation \(f^{-1}(x) = (f(x))^{-1}\) for all \(x \in G\). Using a simple criterion involving the number of elements in G not equal to their own inverse, the classification of finite abelian groups admitting inverse ambiguous functions is achieved. In this paper we aim to extend the results from (2017) to determine the existence of inverse ambiguous functions on members of certain families of non-abelian groups, namely the symmetric groups \(S_n\), the alternating groups \(A_n\), and the general linear groups GL(2, q) over a finite field \(\mathbb {F}_q\).


Inverse function Functional equation Non-abelian groups 

Mathematics Subject Classification

Primary 39B52 Secondary 20B30 


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  1. 1.
    Herzog, M.: Counting elements of order \(p\) Modulo \(p^2\). Proc. Am. Math. Soc. 66(2), 247–250 (1977)zbMATHGoogle Scholar
  2. 2.
    Schmitz, D.: Inverse ambiguous functions on fields. Aequ. Math. 91, 373–389 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.North Central CollegeNapervilleUSA
  2. 2.University of Notre DameSouth BendUSA

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