Finite groups of arbitrary deficiency

The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of relators. Every finite group has non-positive deficiency. We show that every non-positive integer is the deficiency of a finite group – in fact, of a finite $p$-group for every prime $p$. This completes Kotschick’s classification of the integers which are deficiencies of fundamental groups of compact Kaehler manifolds.