Abstract
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.
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Shao, C., Beltrán, A. Coprime action and arithmetical conditions on invariant conjugacy classes. Sci. China Math. 58, 2499–2504 (2015). https://doi.org/10.1007/s11425-015-5007-3
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DOI: https://doi.org/10.1007/s11425-015-5007-3