Because the nilpotent class of Dedekind groups is at best 2, we mainly study nilpotent structures of S(A*, C*)-groups in section 3. Since the locally nilpotent S*(p)-group is nilpotent(see property 3.2),we maily study the locally nilpotent C*(p)-group in the following.
Nonabelian K-theory: The nilpotent class of K1 and general stability
We show that the nilpotent class of any finite group which has only two conjugacy lengths is at most 3.
Let G be a group, one says that G is an S~*(A~*, C~*)-group, if each(abelian, or cyclic)subgroup H of G satisfies that ｜H~G∶H｜<∞; further, one says that G is an S~*(n)(A~*(n), C~*(n))-group, if each(abelian, or cyclic)subgroup H of G satisfies that ｜H~G∶H｜≤n. In this paper, by investigating finite C~*(p)-p-groups the authors have obtained that the nilpotent class of C~*(p)-p-groups is at most 3 and their derived subgroup is elementary abelian p-group.