The Minimum Generating Set Problem

Abstract

Let $G$ be a finite group. In order to determine the smallest cardinality $d(G)$ of a generating set of $G$ and a generating set with this cardinality, one should repeat many times the test whether a subset of $G$ of small cardinality generates $G$. We prove that if a chief series of $G$ is known, then the numbers of these generating tests can be drastically reduced. At most $|G|^{13/5}$ subsets must be tested. This implies that the minimum generating set problem for a finite group $G$ can be solved in polynomial time.