A Condition for Distinguishing Sceneries on Non-abelian Groups

TL;DR

This paper extends a reconstructivity condition from finite abelian groups to non-abelian groups but finds that no non-abelian group random walk satisfies this condition, highlighting fundamental differences.

Contribution

It generalizes the reconstructivity condition to non-abelian groups and demonstrates its limitations in this broader context.

Findings

No non-abelian group random walk satisfies the generalized condition.

The condition is sufficient for abelian groups but not for non-abelian groups.

Highlights fundamental differences between abelian and non-abelian group structures.

Abstract

A scenery $f$ on a finite group $G$ is a function from $G$ to $\{0,1\}$. A random walk $v(t)$ on $G$ is said to be reconstructive if the distributions of 2 sceneries evaluated on the random walk with uniform initial distribution are identical only if one scenery is a shift of the other scenery. Previous results gave a sufficient condition for reconstructivity on finite abelian groups. This paper gives a ready generalization of this sufficient condition to one for reconstructivity on finite non-abelian groups but shows that no random walks on finite non-abelian groups satisfy this sufficient condition.