engleski

Finite 2-groups with exactly one nonmetacyclic maximal subgroup

We determine here the structure of the finite 2-groups with exactly one nonmetacyclic maximal subgroup. All such groups G will be given in terms of generators and relations, and many important subgroups of these groups will be described. Let d(G) be the minimal number of generators of G. We have here that d(G) is equal or less 3 and if d(G)=3, then G' is elementary abelian of order at most 4. Suppose d(G)=2. Then G' is abelian of rank equal or less 2 and G/G' is abelian of type (2, 2^m), m equal or greater 2. If G' has no cyclic subgroup of index 2, then m=2. If G' is noncyclic and G/ F(G') has no normal elementary abelian subgroup of order 8, then G' has a cyclic subgroup of index 2 and m=2. But the most important result is that for all such groups (with d(G)=2) we have G=AB, for suitable cyclic subgroups A and B. Conversely, if G=AB is a finite nonmetacyclic 2-group, where A and B are cyclic, then G has exactly one nonmetacyclic maximal subgroup. Hence, in this paper the nonmetacyclic 2-groups which are products of two cyclic subgroups are completely determined. This solves a long-standing problem studied from 1953 to 1956 by B. Huppert, N. Ito and A. Ohara. Note that if G=AB is a finite p-group, p greater 2, where A and B are cyclic, then G is necessarily metacyclic (Huppert /4/). Hence, we have solved here problem Nr. 776 from Berkovich /1/.

finite 2-group; normal elementary abelian subgroup; nonmetacyclic maximal subgroup; nonmetacyclic minimal nonabelian group

Istraživač na projektu Zvonimir Janko nema matični broj znanstvenika u R. Hrvatskoj. Pripadnik je hrvatske dijaspore u Heidelbergu, Njemačka.