He Guo LIU, Ji Ping ZHANG, Tao XU
Let G be a nilpotent group of finite rank and G = KP, where P is a nilpotent π-group, K is a π'-free normal subgroup, and π does not belong to the spectrum of K. Suppose that1=ζ0G<ζ1G<…<ζCG=G is the upper central series of G, that α and β are two automorphisms of G, that αi and βi are the automorphisms of ζiG/ζi-1G induced by α and β, respectively. Put Ii = Im(αi-βi βiαi), then
(i) in case Ii is a finite cyclic group, and I:= (αβ(g))(βα(g))-1|g ∈ G is a finite subgroup of G, then α, β is a soluble abelian-by-finite group.
(ii) in case Ii is a finite cyclic group, or a divisible group of rank 1, or C ⊕ D, where C is a cyclic group and D is a divisible group of rank 1, or a torsion-free locally cyclic group, or Ii has a normal series 1 < Ji < Ii, one of whose factors is a finite cyclic group and another is torsion-free locally cyclic group, or Ii = D ⊕ Ji, where D is a divisible group of rank 1 and Ji is a torsion-free locally cyclic group, or Ii has a normal series 1 < Ki < Ji < Ii,whose 3 factors are a finite cyclic group, a divisible group of rank 1 and a torsion-free locally cyclic group, respectively, then α, β is a soluble nilpotent-by-abelian-by-finite group.
In particular, if α and β are two π'-automorphisms of G. Then
(iii) in case Ii is a finite cyclic group, and I:= (αβ(g))(βα(g))-1|g ∈ G is a finite subgroup of G, then α, β is an extension of a finite nilpotent π-group by a finite abelian π -group. (iv) in case Ii is a cyclic group, or a divisible group of rank 1, or C ⊕D, where C is a cyclic group and D is a divisible group of rank 1, then α, β is both a soluble residually finite π ∪ π'-group and is an extension of a finitely-generated torsion-free nilpotent group by a finite π ∪ π'-group.
(v) in case Ii is a finite cyclic group, or a divisible group of rank 1, or C ⊕ D, where C is a cyclic group and D is a divisible group of rank 1, or a torsion-free locally cyclic group, or Ii has a normal series 1 < Ji < Ii, one of whose factors is a finite cyclic group and another is a torsion-free locally cyclic group, or Ii = D ⊕ Ji, where D is a divisible group of rank 1 and Ji is a torsion-free locally cyclic group, or Ii has a normal series 1 < Ki < Ji < Ii, whose 3 factors are a finite cyclic group, a divisible group of rank 1 and a torsion-free locally cyclic group, respectively, then α, β is a soluble residually finite π ∪ π'-group and its nilpotent length is at most 4.
In (v), if K is a FC-group, then α, β is an extension of a finitely-generated torsion-free nilpotent group by a finite π ∪ π'-group.
In addition, if G = KP and K is a FC-group, we investigate the dual problem for the lower central series of G and obtain the dual results.