Regular orbits of finite primitive solvable groups
Let \(G\) be a finite primitive solvable irreducible linear group over some finite field. Let \(V\) be the finite vector space on which \(G\) acts naturally. An element \(v\in V\) generates a `regular orbit' if the only element of \(G\) which fixes \(v\) is the identity. The paper proves that unless \(G\) is very uncomplicated there will always be regular orbits of \(G\). More precisely, let us denote by \(F(G)\) the Fitting subgroup of \(G\), and by \(E=[F(G),G,\dots,G]\) the commutator subgroup of \(F(G)\) with \(G\) and the result again with \(G\), with the process repeated until the obtained subgroup ceases to decrease. Then, we denote by \(e=\sqrt{|E/Z(E)|}\), the square root of the order of the quotient of \(E\) by its center. It turns out that \(e\) is always a positive integer, and larger integers roughly correspond to more complicated groups. The following is proved in the paper. If \(e\geq 10\) and \(e\neq 16\), then \(G\) has a regular orbit. Furthermore, it is proved that if \(e\in\{8,9,16\}\), then \(G\) need not have a regular orbit. This strengthens a result of \textit{O. Manz} and \textit{T. R. Wolf} [Representations of solvable groups. Cambridge: Cambridge University Press (1993; Zbl 0928.20008)], where the existence of regular orbits is proved when \(e>118\). For Part II, see [the author, J. Algebra 341, No. 1, 23--34 (2011; Zbl 1247.20021)].
scientific article; zbMATH DE number 5718178