Regular orbits of finite primitive solvable groups. II.
Let \(G\) be a finite solvable group and suppose it acts faithfully, irreducibly and quasi-primitively on a finite vector space \(V\). Then \(G\) has a uniquely determined normal subgroup \(E\) which is a direct product of extraspecial \(p\)-groups for various primes \(p\), and we set \(e=\sqrt{|E/Z(E)|}\). In Part I of the paper [J. Algebra 323, No. 10, 2735-2755 (2010; Zbl 1195.20017)] the author proved that, if \(e\geq 10\) and \(e\neq 16\), then \(G\) has a regular orbit on \(V\). In the present paper, Yang proves that if \(e\in\{5,6,7\}\) then \(G\) also has a regular orbit on \(V\). He also shows that this together with the earlier result completes the list of all \(e\) for which there always is a regular orbit on \(V\). Some further information on the number of regular orbits is also obtained. In addition, some applications of these results are given. For Part III, see [the author et al., J. Algebra 590, 139-154 (2022; Zbl 07420015)].
scientific article; zbMATH DE number 6005424