LAUSR

Let G be a group with an involution $$x \mapsto x^*$$ x ↦ x ∗ , let $$\mu :G \rightarrow \mathbb {C}$$ μ : G → C be a multiplicative function such that $$\mu (xx^*) = 1$$ μ ( x x ∗ ) = 1 for all $$x \in G$$ x ∈ G , and let the pair $$f,g:G \rightarrow \mathbb {C}$$ f , g : G → C satisfy that $$\begin{aligned} f(xy) + \mu (y)f(xy^*) = 2f(x)g(y), \ \forall x,y \in G. \end{aligned}$$ f ( x y ) + μ ( y ) f ( x y ∗ ) = 2 f ( x ) g ( y ) , ∀ x , y ∈ G . For G compact we obtain: If g is abelian, then f is abelian. For G nilpotent we obtain: (1) If G is generated by its squares and $$f \ne 0$$ f ≠ 0 , then g is abelian. (2) If g is abelian, but not a multiplicative function, then f is abelian.