Finite groups with regular orbits on vector spaces
Let \(\pi\) '(A) be the set of all primes which not divide the order of the finite group A, \(p\in \pi'(A)\), \(V_ p(A):=\) all \(F_ qA\)-modules for arbitrary p-power q. \(S_ p(A):=\{V\in V_ p(A)|\) A has a regular orbit on \(V\}\). \(\tau(A)= \{p\in \pi '(A)|\) \(V_ p(A)=S_ p(A)\}\), \(\sigma (A)=\cap_{{\mathcal U}\leq A}\tau ({\mathcal U}).\) Theorem 1. If P is a p-group, \(p>2\), then \(\pi\) '(P)-\(\sigma\) (P)\(\subseteq \{2\}\). Theorem 2. Let \(\hat M\), \(\hat F\) denote Mersenne resp. Fermat primes. If P is a 2-group then \(\pi\) '(P)-\(\sigma\) (P)\(\subseteq \hat M\cup \hat F\). It is given some result on the connection between \(V_ p(A)\), \(S_ p(A)\), \(p\in \pi '(A)\).
scientific article; zbMATH DE number 3975187