2016-11-15T19:40:59

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Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on a finite vector space V . Then G has a uniquely determined normal subgroup E which is a direct product of extraspecial p-groups for various p and we denote e = √ |E/Z(E)|. We prove that when e 10 and e = 16, G will have at least 5 regular orbits on V . We also construct groups with no regular orbits on V when e = 8, 9 and 16.

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Abstract. We prove that if G is a finite solvable group and 3 |G : F(G)|, then the index of the Fitting subgroup of G is at most the square of the largest irreducible character degree of G.

The number of different prime divisors of the order of a finite group 𝐺 is bounded above by a quartic function of the maximum number of different prime divisors of the order of a single element in 𝐺. This improves earlier results of J. Zhang,