Quadratic automorphisms of abelian groups

Abstract

We study automorphism groups of Abelian groups G generated by quadratic automorphisms, that is, those of which each being an element of the endomorphism ring of G is a root of the quadratic equation x2 + αx + β·1 with integral coefficients. Quadratic automorphisms are most notably exemplified by elements of orders 3 and 4 in groups of regular automorphisms: these are roots of the equations x2 + x + 1 and x2 + 1. respectively. Let A be generated by two quadratic automorphisms a and b of an Abelian group G. Then the following statements hold: (1) if the exponent m of G and the order n of ab are finite then A is a finite group of order at most m2n − 1; (2) if A is periodic then it is finite. Moreover, both of the finite conditions in (1) are essential. A consequence of these results is obtaining a description of periodic groups of regular automorphisms. generated by two automorphisms whose orders do not exceed 4.