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2-group 2-subgroups of G action of G assume Aut(N automorphism group character of G character values contains a primitive contains all elements contradiction Corollary cyclic group cyclic of odd cyclic Sylow denote dimV(g divide the order divides n/m eigenspace elements of order finite field finite field F fixed element Frobenius action G are cyclic g e G g in G Galois conjugate Galois group Galois theory group of G hypothesis of Theorem Irr(G irreducible constituents k-eigenvalue property K:QJ Lemma Let G minimal normal subgroup normal in G normal q-complement odd order order of G order q p-solvable positive integer prime divisor primitive q root Proof of Step proof of Theorem proper factor group prove Theorem quaternion group root of unity solvable Frobenius complement solvable group subgroup of order Sylow 2-subgroups unique cyclic subgroup unique minimal normal unless q values in Q vector space write dimV y-eigenspace of g