On Certain Automorphisms of Nilpotent Groups

Isomorphism between automorphism groups of finitely generated groups

“…For convenience, we denote γ k (G)-pointwise inner automorphism of G by Aut k−pwi (G). As another application of Lemma 2.1, we get the following two results of Azhdari [3]. The second one generalizes Theorem 2.2(i) of [3].…”

“…As another application of Lemma 2.1, we get the following two results of Azhdari [3]. The second one generalizes Theorem 2.2(i) of [3].…”

“…Azhdari and Malayeri [4, Theorem 0.1] (see also [5,Theorem 2.3] for correct version) generalized this result of Attar and proved that if G is a finitely generated nilpotent group of class 2, then C * ≃ Inn(G) if and only if Z(G) is infinite cyclic or Z(G) ≃ C m × H × Z r , where C m ≃ p∈π(G/Z(G)) Z(G) p , H ≃ p / ∈π(G/Z(G)) Z(G) p , r ≥ 0 is the torsion-free rank of Z(G) and G/Z(G) is of finite exponent dividing m. We prove a technical lemma, Lemma 2.1, and as a consequence give a short and easy proof of this main theorem of Azhdari and Malayeri. We also obtain short and alternate proofs of Corollary 2.1 of [5], and Propostion 1.11 and Theorem 2.2(i) of [3]. Some other related results for finitely generated and finite p-groups are also obtained.…”

“…, where c is the torsion-free rank of Z(G). In 1911, Burnside [6, Note B, p. 463] gave the notion of pointwise inner automorphism of a group G. An automorphism α of G is called pointwise inner automorphism of G if x and α(x) are conjugate for each x ∈ G. Let H be a characteristic subgroup of G. As defined in [3], an automorphism α of G is called H-pointwise inner if for each element x ∈ G, there exists h ∈ H such that α(x) = x h = x[x, h]. For convenience, we denote γ k (G)-pointwise inner automorphism of G by Aut k−pwi (G).…”