Subdirectly Reducible Groups and Edge-minimal Graphs with given Automorphism Group

A group is subdirectly reducible if it has two non-trivial normal subgroups with trivial intersection. Such groups may be an easy case in certain inductive arguments. We prove that every solvable finite group can be generated by at most one subdirectly reducible subgroup together with two subgroups of prime-power order and one element. We also prove that every group of prime-power order can be generated by at most one subdirectly reducible subgroup together with two elements. Combining these two results with the fact that every finite group is generated by two conjugate solvable subgroups [1], we have a tool for certain inductive proofs about arbitrary finite groups. As an application we prove that any function on finite groups satisfying three simple properties (bounded on cyclic groups, additive when passing from subgroups to the group they generate, and sufficiently 'well-behaved' on subdirectly reducible groups) is necessarily bounded. Finally we apply this last result to confirm a conjecture of Babai (1981) on edge-minimal graphs with given automorphism group: for every finite group G there exists a graph X (and a lattice L) whose full automorphism group is isomorphic to G such that X has at most c|<7| edges (and L has at most 2c\G\ elements) for some constant c < 300.