INTERPLAY BETWEEN THE ALGEBRAIC STRUCTURE OF A GROUP AND ARITHMETIC PROPERTIES OF ITS SPECTRUM

Abstract

We investigate the interplay between the algebraic structure of a group G and arithmetic properties of its spectrum σ(G) which consists of the eigenvalues of all the inner automorphisms of G. A complex number λ is called an eigenvalue of a group automorphism A:G→G if φ◦A|H=λ·φ for some non-trivial homomorphism [Formula: see text] defined on an A-invariant subgroup H⊂G. It is shown that many properties of a group G (such as the presence of a finitely generated subgroup of infinite rank, nilpotence, periodicity, polycyclicity, etc.) are coded in its spectrum. In this paper the spectra are applied to the investigation of the so-called reversive properties of groups. The paper ends with a list of related open problems.