Finite Abelian Groups with the Strong Endomorphism Kernel Property
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Finite Abelian Groups with the Strong Endomorphism Kernel Property Jie FANG
Zhong Ju SUN1)
School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665, P. R. China E-mail : [email protected] [email protected] Abstract An endomorphism h of a group G is said to be strong whenever for every congruence θ on G, (x, y) ∈ θ implies (h(x), h(y)) ∈ θ for every x, y ∈ G. A group G is said to have the strong endomorphism kernel property if every congruence on G is the kernel of a strong endomorphism. In this note, we study the strong endomorphism kernel property in the class of Abelian groups. In particular, we show that a finite Abelian group has the strong endomorphism kernel property if and only if it is cyclic. Keywords
Strong endomorphism kernel property, congruence, Abelian group, cyclic group
MR(2010) Subject Classification
1
08A35, 20K30
Introduction
A notion of the (strong) endomorphism kernel property was first introduced by Blyth, Fang and Silva in the context of Ockham algebras and MS-algebras (see [1, 2]). This universal algebraic concept has been investigated in several classes of lattice-ordered algebras. For example, Blyth and Silva gave a complete description of the structure of those MS-algebras that have the strong endomorphism kernel property by way of Priestley topological duality [2]. Similar approaches are adopted in considering this concept in the context of distributive p-algebras [4] and double p-algebras [3], and in that of double MS-algebras [5]. More classes of lattice-ordered algebras with the strong endomorphism kernel property can be found in [6–9]. If A is an algebra, then an endomorphism h on A is said to be strong whenever for every congruence θ on A, (∀x, y ∈ A) (x, y) ∈ θ =⇒ (h(x), h(y)) ∈ θ. An algebra A is said to have the strong endomorphism kernel property if every congruence on A, other than the universal congruence ιA = A × A, is the kernel of a strong endomorphism of A. The exception for the universal congruence ιA appears in above definition with the purpose that algebras with two or more nullary operations have the condition of the endomorphism kernel. It is actually not necessary for the singleton algebras or algebras with at most one nullary operation, such as groups. With this consideration, we shall say that a group G has the Received October 10, 2019, revised April 6, 2020, accepted May 7, 2020 1) Corresponding author
Finite Abelian Groups with the Strong Endomorphism Kernel Property
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strong endomorphism kernel property (SEKP, for short) if every congruence on G is the kernel of a strong endomorphism. In this note, we shall particularly consider those Abelian groups that have the SEKP. We recall from [10] that a congruence θ on a group (G, ·) is an equivalence relation such that (x, y) ∈ θ implies that (x−1 , y −1 ) ∈ θ and that for every z ∈ G, (x · z, y · z) ∈ θ and (z · x, z · y) ∈ θ. It is well known (see [10]) that
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