摘要
Let A and G be finite groups and suppose that A acts coprimely on G via automorphisms. We show that if 4 divides no A-invariant conjugacy class size of G, then G is solvable. We also characterize the A-invariant structure of G under certain arithmetical conditions on the set of A-invariant class sizes of G by means of the fixed point subgroup, some of which imply the solvability of G. Thus, we extend, for coprime action, several results appeared in the literature on class sizes.
Let A and G be finite groups and suppose that A acts coprimely on G via automorphisms. We show that if 4 divides no A-invariant conjugacy class size of G, then G is solvable. We also characterize the A-invariant structure of G under certain arithmetical conditions on the set of A-invariant class sizes of G by means of the fixed point subgroup, some of which imply the solvability of G. Thus, we extend, for coprime action, several results appeared in the literature on class sizes.
基金
supported by National Natural Science Foundation of China(Grant No.11301218)
the Nature Science Fund of Shandong Province(Grant No.ZR2014AM020)
University of Jinan Research Funds for Doctors(Grant Nos.XBS1335 and XBS1336)
the Valencian Government
Proyecto PROMETEO/2011/30
the Spanish Government
Proyecto(Grant No.MTM2010-19938-C03-02)
