Let G be a linear group acting on the finite vector space V and assume that (|G|, |V |) = 1. In this paper we prove that G has a base size at most two and this estimate is sharp. This generalizes and strengthens several former results concerning base sizes of coprime linear groups. As a direct consequence, we answer a question of I. M. Isaacs in the affirmative. Via large orbits this is related to the k(GV ) theorem.

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Abstract. The flow semigroup, introduced by John Rhodes, is an invariant for digraphs and a complete invariant for graphs. After collecting together previous partial results, we refine and prove Rhodes's conjecture on the structure of the maximal groups in the flow semigroup for finite, antisymmetric, strongly connected digraphs.Building on this result, we investigate and fully describe the structure and actions of the maximal subgroups of the flow semigroup acting on all but k points for all finite digraphs and graphs for all k ≥ 1. A linear algorithm (in the number of edges) is presented to determine these so-called 'defect k groups' for any finite (di)graph.Finally, we prove that the complexity of the flow semigroup of a 2-vertex connected (and strongly connected di)graph with n vertices is n − 2, completely confirming Rhodes's conjecture for such (di)graphs.

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