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- Title
On nilpotency of the group of outer class-preserving automorphisms of a group.
- Authors
Rai, Pradeep K.
- Abstract
Let G be a group and Zj(G), for j ≥ 0, be the jth term in the upper central series of G. We prove that if c(G/Zj(G)), the group of outer class-preserving automorphisms of G/Zj(G), is nilpotent of class k, then c(G) is nilpotent of class at most j + k. Moreover, if c(G/Zj(G)) is a trivial group, then c(G) is nilpotent of class at most j. As an application we prove that if γi(G)/γi(G) ∩ Zj(G) is cyclic then c(G) is nilpotent of class at most i + j, where γi(G), for i ≥ 1, denotes the ith term in the lower central series of G. This extends an earlier work of the author, where this assertion was proved for j = 0. We also improve bound on the nilpotency class of c(G) for some classes of nilpotent groups G.
- Subjects
GROUP theory; SET theory; AUTOMORPHISMS; MATHEMATICAL proofs; NILPOTENT groups
- Publication
Journal of Algebra & Its Applications, 2016, Vol 15, Issue 2, p-1
- ISSN
0219-4988
- Publication type
Article
- DOI
10.1142/S0219498816500262