The absolute center of finite groups

Abstract

In this paper we first investigate the relationship between the absolute center

A Note on Absolute Central Automorphisms of Finite $p$-Groups

“…Then we define θ = θ f : G → G by g θ = gf (g). Since by [11,Corollary 3.7], g −1 g θ ∈ L(G) ≤ Φ(G), for every element g ∈ G, we may write G as the product of the image of θ and the Frattini subgroup of G and so the image of θ must be G itself. Hence θ is an automorphism of G.…”

“…In [11], Meng and Guo proved that for a finite group G, if C 2 is not a direct factor of G, then L(G) ≤ Φ(G). We end this section by characterizing the finite non-abelian p-groups G with cyclic Frattini subgroup for which Aut L (G) = Aut Φ (G).…”

“…Then by Corollary 3.5, Aut L (G) = Inn(G) if and only if p = 2. In the following corollaries, we use Theorems 4.7 and 5.1 of [11] and classify all finite p-groups G of order p n (4 ≤ n ≤ 5), such that Aut L (G) = Inn(G). First we recall the following concept, which was introduced by Hall in [6].…”

“…Then G/Z(G) ∼ = C 2 p and so d(G/L(G)) > 1. We observe that G = L(G) ∼ = C p and Z(G) = Φ(G), by using Theorems 2.2, 3.2, [3, Theorem 2.2] and [11,Theorem 5.1]. So d(G) = 2 and by [16], G is a minimal non-abelian p-group.…”

“…p 2 , then by [16], [11,Theorem 4.7], a contradiction. Finally, assume that Hence by [16], G is a minimal non-abelian 2-group.…”