Let G be a finite group and Aut(G) be the group of automorphisms of G. Then, the autocentralizer of an automorphism α ∈ Aut(G) in G is defined as CG(α) = {g ∈ G|α(g) = g}. Let Acent(G) = {CG(α)|α ∈ Aut(G)}. If |Acent(G)| = n, then G is an n-autocentralizer group. In this paper, we classify all n-autocentralizer abelian groups for n = 6, 7 and 8. We also obtain a lower bound on the number of autocentralizer subgroups for p-groups, where p is a prime number. We show that if p ̸ = 2, there is no n-autocentralizer p-group for n = 6, 7. Moreover, if p = 2, then there is no 6-autocentralizer p-group.
Let G be a finite non-abelian p-group, where p is a prime number, and Aut(G) be the group of all automorphisms of $G$. An automorphism alpha of $G$ is called absolute central automorphism if, x^{-1}alpha(x) lies in L(G), where L(G) is the absolute center of G. In addition, alpha is an absolute Frattini automorphism if x^{-1}alpha(x) is in Phi(L(G)), where Phi(L(G)) is the Frattini subgroup of the absolute center of G, and let LF(G) denote the group of all such automorphisms of G. Also, we denote by C_{LF(G)}(Z(G)) and C_{LA(G)}(Z(G)), respectively, the group of all absolute Frattini automorphisms and the group of all absolute central automorphisms of G, fixing elementwise the center Z(G) of G . We give necessary and sufficient conditions on a finite non-abelian p-group G of class two such that C_{LF(G)}(Z(G))=C_{LA(G)}(Z(G)). Moreover, we investigate the conditions under which LF(G) is a torsion-free abelian group.
Let G be a group and let M be a characteristic subgroup of G. We denote by Aut M M (G) the set of all automorphisms of G which centralize G/M and M. In this paper, we give necessary and sufficient conditions for the equality of Aut M M (G) with Aut M (G) and C Aut M M (G) (Z(G)).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.