Invariant class sizes and solvability of finite groups under coprime action. (Q2792248)

Suppose \(A\) and \(G\) are finite groups of relatively prime orders such that \(A\) acts on \(G\) via automorphisms, where \(A\) acts also on the set of conjugacy classes of \(G\). Several arithmetical properties on the sizes of certain classes which are left fixed by \(A\) and how they reflect on the \(A\)-invariant structure and may imply the solvability of \(G\), are studied in the paper. As such, one obtains, for instance:NEWLINENEWLINE a) if the size of every \(A\)-invariant conjugacy class of an odd power element of \(G\) is a 2-power or not divisible by 4, then \(G\) is solvable;NEWLINENEWLINE b) if the size of every \(A\)-invariant conjugacy class of 2-elements of \(G\) is a prime power, then \(G\) is solvable;NEWLINENEWLINE c) a fixed prime \(p\) divides no \(A\)-invariant class size of elements of prime power order of \(G\) if and only if \(G\) has an \(A\)-invariant Sylow \(p\)-subgroup centralized by \(C_G(A)\);NEWLINENEWLINE d) if the sizes of the \(A\)-invariant classes of prime power order elements of \(G\) are divisible by at most two fixed primes, then \(G\) is solvable.NEWLINENEWLINE As it is to be expected, the proofs of the statements are (very) technical and do depend on the classification of the finite simple groups.