Frobenius Groups Generated by Quadratic Elements

An automorphism $a$ of a group $X$ is said to be quadratic if there exist integers $m=m(a)$ and $n=n(a)$ such that $x^{a^2}=x^n(x^m)^a= x^nx^{ma}$ for any $x\in X$. If $G$ is a Frobenius group then an element $g\in G$ is said to be quadratic if $g$ induces, by conjugation in the core of $G$, a quadratic automorphism. By definition, a group $H$ acts on a group $F$ freely if $f^h=f$ for $f\in F$ and $h\in H$ only with $f=1$ or $h=1$. It is proved that a Frobenius group generated by two quadratic elements is finite and its core is commutative. In particular, any Frobenius group generated by two elements of order at most 4 is finite. Also we argue that a Frobenius group with finitely generated soluble core is finite. The results mentioned are used to show that a group $G$ acting freely on an Abelian group is finite if it is generated by elements of order 3, and the order of a product of every two elements of order 3 in $G$ is finite.