The cyclic length of ap-group need not be logarithmic

Abstract

The cyclic length l(G) of a finite supersolvable group G is defined to be the smallest number l such that there is a series G = N o > N I > . . . > N ~ = I , where No, N1 . . . . . Nl are l+1 different normal subgroups of G and all factor groups Ni_l/Ni (i=1 . . . . . l) are cyclic. L. R6dei asked the question whether or not the cyclic lengh has a logarithmic property on the class of p-groups: i.e. does l(G1 • G2) = =l(G1)+l(G2) hold, whenever Gx, G2 are p-groups for a certain prime p? At the first glance the answer could be expected to be in the positive, especially in view of the validity on the class of abelian p-groups. In the present note we will bring this expectation to nought not only by means of a single counterexample but more generally by assigning a p-group G to each abelian p-group H so that l ( G • < l ( G ) + l ( H ) takes place. The notation will be standard and can be found in [1]. All groups under consideration are finite.