On quadratic automorphisms of abelian groups

A. Kh. Zhurtov

Algebra i logika, Tome 39 (2000) no. 3, pp. 320-328.

Résumé

We study subgroups of automorphism groups of abelian groups $G$ generated by quadratic automorphisms (an automorphism is said to be quadratic if, as an element of the endomorphism ring of $G$, it is a root of the quadratic equation $x^2+\alpha x+\beta\cdot1$ with integer coefficients). The most important examples of quadratic automorphisms are elements of orders 3 and 4 in groups of regular automorphisms: they are roots of the equations $x^2+x+1$ and $x^2+1$, respectively. Suppose that the group $A$ is generated by two quadratic automorphisms $a$ and $b$ of an abelian group $G$. Then the following statements hold: (1) if the period $m$ of $G$ and the order $n$ of the product $ab$ are finite, then $A$ is a finite group of order at most $m^{2n}-1$; (2) if $A$ is a periodic group, then it is finite. Moreover, we prove that both finiteness conditions in (1) are essential. As a consequence of these results, we obtain a description of the periodic groups of regular automorphisms generated by two automorphisms of order at most 4.