Let $G$ be an arbitrary group. We define a gain-line graph for a gain graph $(��,��)$ through the choice of an incidence $G$-phase matrix inducing $��$. We prove that the switching equivalence class of the gain function on the line graph $L(��)$ does not change if one chooses a different $G$-phase inducing $��$ or a different representative of the switching equivalence class of $��$. In this way, we generalize to any group some results proven by N. Reff in the abelian case. The investigation of the orbits of some natural actions of $G$ on the set $\mathcal H_��$ of $G$-phases of $��$ allows us to characterize gain functions on $��$, gain functions on $L(��)$, their switching equivalence classes and their balance property. The use of group algebra valued matrices plays a fundamental role and, together with the matrix Fourier transform, allows us to represent a gain graph with Hermitian matrices and to perform spectral computations. Our spectral results also provide some necessary conditions for a gain graph to be a gain-line graph.

28 pages, 6 figures, 1 table