English

Inverse ambiguous maps on infinite groups

Group Theory 2026-08-06 v1

Abstract

Let GG be a group. A bijection f ⁣:GGf\colon G\to G is called inverse ambiguous if f1(x)=f(x)1f^{-1}(x)=f(x)^{-1} for every xGx\in G. We prove that every infinite group admits an inverse ambiguous function. An inverse ambiguous automorphism, however, can occur only on an abelian group. For a finitely generated abelian group AZrTA\cong\Z^r\oplus T, with TT finite, we show that AA admits an inverse ambiguous automorphism if and only if rr is even and TT admits one. This leads to an explicit classification by combining the result with Toborg's finite classification. We also prove that an infinite locally finite group is abelian if each of its proper subgroups admits an inverse ambiguous automorphism, and we give examples showing that both hypotheses are necessary. Finally, we prove that a residually finite group is abelian whenever all its finite quotients admit inverse ambiguous automorphisms. We also construct an infinite residually finite abelian group whose finite quotients all admit inverse ambiguous automorphisms, although the group itself admits none.

Cite

@article{arxiv.2608.06191,
  title  = {Inverse ambiguous maps on infinite groups},
  author = {Sezen Bostan and Kıvanç Ersoy},
  journal= {arXiv preprint arXiv:2608.06191},
  year   = {2026}
}