Counting generators in groups that are generated by two subgroups of prime power order.

摘要

Let P be a d-generated p-group and Q be a d-generated q-group for distinct primes p and q. It has been conjectured that for any finite group G = ⟨P, Q⟩, G is (d + 1)-generated. Lucchini determined that any minimal counterexample to this conjecture embeds into Lt where L has a unique minimal normal subgroup M = Sn with S nonabelian simple. Up to information on finite simple groups, we prove that L/M is (d + 1)-generated or nonsolvable.