In this paper, we introduce and study a new class of centralizers which are called triple θ-centralizers. We will see that the notions of triple θ-centralizers and θ-centralizers are different. Some observations concerning triple θ-centralizers and approximate triple θ-centralizers are given.

This is a survey article on centralizers of finite subgroups in locally finite, simple groups or LFS-groups as we will call them. We mention some of the open problems about centralizers of subgroups in LFS-groups and applications of the known information about the centralizers of subgroups to the structure of the locally finite group. We also prove the following: Let G be a countably in finite nonlinear LFS-group with a Kegel sequence K={(Gi, Ni) | iÎN}. If there exists an upper bound for {|Ni| | iÎN}, then for any finite semisimple subgroup F in G the subgroup CG (F) has elements of order pi for in finitely many distinct prime pi. In particular CG (F) is an in finite group. This answers Hartley’s question provided that there exists a bound on {|Ni| | iÎN}.

For any group G, let C (G) denote the set of centralizers of G. We say that a group G has n centralizers (G is a Cn-group) if |C (G)|=n. In this note, we prove that every finite Cn-group with n£21 is soluble and this estimate is sharp. Moreover, we prove that every finite Cn-group with |G|<30n+15/19 is non-nilpotent soluble. This result gives a partial answer to a conjecture raised by A. Ashrafi in 2000.

Let A be a Banach algebra with unity 1, and θ: A → A be an continuous automorphism. In this paper we characterize a continuous linear map T: A → A which satisfies one of the following conditions: a, b ∈ A, ab = w =⇒ θ(a)T(b) = T(w), a, b ∈ A, ab = w =⇒ T(a)θ(b) = T(w), or a, b ∈ A, ab = w =⇒ θ(a)T(b) = T(a)θ(b) = T(w), where w 6= 0 is a left (right) separating point of A.

Let G be a group. For an element a 2 G, denote by C (a) the second centralizer of a in G, which is the set of all elements b 2 G such that bx = xb for every x 2 G that commutes with a. Let M be any maximal abelian subgroup of G. Then C 2 2 (a) ,M for every a 2 M. The abelian rank (a-rank) of M is the minimum cardinality of a set A ,M such that ∪,2 C a 2 A (a) generates M. Denote by Sn the symmetric group of permutations on the set X = f 1, : : :, n g. The aim of this paper is to determine the maximal abelian subgroups of Sn of a-rank 1 and describe a class of maximal abelian subgroups of Sn of a-rank at most 2.

Let $R$ be a $ 2$-torsion-free semiprime ring and $\theta$ be an epimorphism of $R$ . In this paper , under special hypotheses , we prove that if $T : R\longrightarrow R$ is an additive mapping such that $ $ T(xyx)=θ(x)T(y)θ(x) , $ $ holds for all $x , y\in R$ , then $T$ is a $θ$-centralizer either $R$ is unital or $θ(Z(R))=Z(R)$.