Abstract
For a group G, D(G) denotes the group of all derival automorphisms of G. For a finite nilpotent group of class 2, it is shown that D(G)≅Hom(G/γ2(G),γ2(G)). We prove that if G is a nilpotent group of class ≥3 such that Z(G)⊆γ2(G) and D(G/Z(G))=Inn(G/Z(G)), then D(G)=Inn(G) if and only if Autcent(G)=Z(Inn(G)). Finally, for an odd prime p, we classify all p-groups of order pn,1≤n≤5, for which D(G)=Inn(G).
Highlights
We prove that if G is a nilpotent group of class ≥ 3 such that Z(G) ⊆ γ2(G) and D(G∕Z(G)) = Inn(G∕Z(G)), D(G) = Inn(G) if and only if Autcent(G) = Z(Inn(G))
Note that the inner automorphism Ta:G ⟶ G given by Ta(x) = a−1xa, for all x ∈ G, is a particular example of a class-preserving automorphism
The group of all inner automorphisms of G is denoted by Inn(G) and it is a normal subgroup of Autc(G)
Summary
Group of all class-preserving automorphisms is denoted by Autc(G). The group of all inner automorphisms of G is denoted by Inn(G) and it is a normal subgroup of Autc(G). The set of all central automorphisms of G is a normal subgroup of Aut(G) and it is denoted by Autcent(G). It has been shown by Sah (1968) that Autcent(G) = CAut(G)(Inn(G)). Ghoraishi (2015) find out the necessary and sufficient condition for equality of class preserving and central automorphism of a finite group. In another note Yadav (2008) studied class-preserving automorphisms of group of order p5, p an odd prime and proved that Autc(G) = Inn(G) for all groups G of order p5 except two isoclinism families.
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