Volume 13, Issue 2 (7-2013)
2013, 13(2): 285-294 |
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Abstract: (4937 Views)
An automorphism $theta$ of a group $G$ is pointwise inner if$theta(x)$ is conjugate to $x$ for any $xin G$. It is interesting and natural to discuss the question of ``finding necessary and sufficient conditions for a group $G$ such that certain subgroups of $text{Aut}(G)$ be equal''.
There are some well-known results in this regard for finite groups.
In this paper, we find a necessary and sufficient condition in certain finitely generatednilpotent groups of class 2 for which $mathrm{Aut}_{pwi}(G)simeq mathrm{Inn}(G)$. We also prove that
in a nilpotent group of class 2 with cyclic commutator subgroup $mathrm{Aut}_{pwi}(G)simeq mathrm{Inn}(G)$ and the quotient$mathrm{Aut}_{pwi}(G)/mathrm{Inn}(G)$ is torsion. In particular if $G'$ is a finite cyclic group then $mathrm{Aut}_{pwi}(G)= mathrm{Inn}(G)$.
There are some well-known results in this regard for finite groups.
In this paper, we find a necessary and sufficient condition in certain finitely generatednilpotent groups of class 2 for which $mathrm{Aut}_{pwi}(G)simeq mathrm{Inn}(G)$. We also prove that
in a nilpotent group of class 2 with cyclic commutator subgroup $mathrm{Aut}_{pwi}(G)simeq mathrm{Inn}(G)$ and the quotient$mathrm{Aut}_{pwi}(G)/mathrm{Inn}(G)$ is torsion. In particular if $G'$ is a finite cyclic group then $mathrm{Aut}_{pwi}(G)= mathrm{Inn}(G)$.
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