On a combinatorial problem in group theory

Abstract

In this paper we continue the study of p-groups G of square order p 2 n and investigate the existence of partial congruence partitions (sets of mutually disjoint subgroups of order p n ) in G. Partial congruence partitions are used to construct translation nets and partial difference sets, two objects studied extensively in finite geometries and combinatorics. We prove that the maximal number of mutually disjoint subgroups of order p n in a group G of order p 2 n cannot be more than ( p n−1 −1)( p−1) −1 provided that n ⩾ 4 and that G is not elementary abelian. This improves an earlier result (D. Hachenberger, J. Algebra 152, 1992, 207–229) and as we do not distinguish the cases p = 2 and p odd in the present paper, we also have a generalization of D. Frohardt's theorem on 2-groups ( J. Algebra 107, 1987, 153–159). Furthermore we study groups of order p 6. We can show that for each odd prime number, there exist exactly four nonisomorphic groups which contain at least p + 2 mutually disjoint subgroups or order p 3. Again, as we do not distinguish between the even and the odd case in advance, we in particular obtain the classification of groups of order 64 which contain at least 4 mutually disjoint subgroups of order 8 found by D. Gluck ( J. Combin. Theory Ser. A 51, 1989, 138–141) and A. P. Sprague ( Mitt. Math. Sem. Giessen 157, 1982, 46–68).