Orbit sizes of p-groups
When does a p-group P, acting faithfully and coprimely on a solvable group H, have a long orbit (preferably a regular orbit) in this action? Around this question [see \textit{D. Passman}, Trans. Am. Math. Soc. 123, 99-111 (1966; Zbl 0139.018), or \textit{T. Berger}, J. Algebra 51, 416-424 (1978; Zbl 0377.20007), or \textit{P. Fleischmann}, J. Algebra 103, 211-215 (1986; Zbl 0604.20014), for instance] several authors have found results. Here some general and easy ideas are proposed, relying on a paper of \textit{R. Gow} [J. Algebra 65, 421-426 (1980; Zbl 0445.20003)]. There are four lemmas and three theorems in the paper under review, not new, but proved by means of the ideas mentioned before. We state two of the theorems. Theorem 4. Let the non-trivial finite p- group P act irreducibly and faithfully on the finite dimensional vector space V over \({\mathbb{F}}_ q\), q prime, \(q\neq p\). Then P induces at least two regular orbits on V, whenever the ordered pair \(\{\) p,q\(\}\) does not belong to the set \(\{\) \(\{\) 2, Mersenne prime\(\}\), \(\{\) 2, Fermat prime\(\}\), \(\{\) Fermat prime, \(2\}\) \(\}\). Theorem 8. Suppose that the finite p-group P contains no section isomorphic to \(C_ p wr C_ p\). Then P has a regular orbit in its action on any finite dimensional vector space V over \({\mathbb{F}}_ q\), where \(p\neq q\), q prime.
scientific article; zbMATH DE number 4146230
scientific article