Endomorphism Near-ring Research Articles

Group object quandles and their endomorphism nearrings

Group object quandles are sets with compatible group and quandle operations. Their endomorphisms will themselves be group objects and function composition becomes a third compatible operation, giving the structure of a nearring, a ring-like structure lacking one distributive property. We give an orbit decomposition of group object quandles in terms of cosets of a certain subgroup of the additive group, and describe them as central extensions of this subgroup. We then describe and count the elements of the endomorphism nearring and study its ideal structure.

QUASI-INJECTIVE NEAR-RING GROUPS

We extend the concepts of quasi-injective modules and their en- domorphism rings to near-ring groups. We attempt to derive the near-ring character of the set of endomorphism of quasi-injective N-groups under certain conditions and this leads us to a near-ring group structure which motivates us to study various characteristics of the structure. If E is a quasi-injective N-group and S = End(injectivehull of E) then we study the structure ES and various properties of ES. It is proved that ES is a minimal quasi-injective exten- sion of E and any two minimal quasi-injective extensions are equivalent. This structure motivates to study the Jacobson radical of endomorphism near-ring of quasi-injective N-group E. It is established that the near-ring modulo the Jacobson radical is a regular near-ring. Some properties of quasi-injective N- groups relating essentially closed N-subgroups and complement N-subgroups are established.

E-groups do not have proper semidirect sum decompositions

An E-group is a group in which each element commutes with any of its endomorphic images. Hence E-groups provide a generalization of abelian groups. It is easy to show that the endomorphism near-ring (the group generated additively by the endomorphisms) of an E-group is a ring, just as the endomorphism near-ring of an abelian group is a ring. In this paper, we establish the fact that, like abelian groups, E-groups have no proper semidirect sum decompositions (i.e. a semidirect sum decomposition of an E-group must be a direct sum decomposition).

An interesting orbit-induced partition of the class of all groups

Given that the set of endomorphisms of a group is contained in the set of distributive elements of its endomorphism near-ring, which, in turn, is contained in the endomorphism near-ring, we show that the class of all groups is partitioned into four nonempty subclasses when all combinations of these inclusions, proper or non-proper, are considered. Furthermore, a characterization of each subclass is given in terms of the orbits of the underlying group.

Polynomial Functions and Endomorphism Near-Rings on Certain Linear Groups

We describe the unary polynomial functions on the non-solvable groups G with SL(n, q) ≤ G ≤ GL(n, q) and on their quotients G/Y with Y ≤ Z(G), and we compute the size of the inner automorphism near-ring I(G/Y). We compare this near-ring to the endomorphism near-ring E(G/Y), and we obtain a full characterization of those G and Y for which I(G/Y) = E(G/Y) holds. For the case Y = {1}, this characterization yields that we have E(G) = I(G) if and only if G = SL(n, q). We investigate the automorphism near-ring A(G), and we show that for all non-solvable groups G with SL(n, q) ≤ G ≤ GL(n, q), we have I(G) = A(G). Our results are based on a description of the polynomial functions on those non-abelian finite groups G that satisfy the following conditions: G′ = G″, G/Z(G) is centerless, and there is no normal subgroup N of G with G′ ∩ Z(G) < N < G′.

Split Metacyclic p-Groups That Are A-E Groups

A group G is an A-E group if the endomorphism nearring of G generated by its automorphisms equals the endomorphism nearring generated by its endomorphisms. In this paper we set out to determine those p-groups G that are semidirect products of cyclic groups and are A-E groups. We show that no such groups exist when p = 2. When p is odd, we show that G is an A-E group whenever the nilpotency class of G is less than p. Examples are given to show no conclusion can be drawn when the nilpotency class is greater than or equal to p.

On semi-endomorphisms of groups

Given a group G, a mapping αG→G is said to be a semi-endomorphism of G if for all x,y ∈ G. It is shown that any non-trivial zero preserving semi-endomorphism of a finite simple group of order greater than two is either an automorphism or an anti-automorphism. Moreover, the semi-endomorphisms of S n, the symmetric group of degree n n ≥ 4, are described. As an application, it is proved that the semi-endomorphism nearring S(Sn ) of Sn with n ≥ 3 is equal to E(Sn ) + Mc (Sn ) where E(Sn )is the endomorphism nearring of Sn , and Mc (S n) is the nearring of constant mappings of Sn .

The semidirect products of finite cyclic groups that areI-E groups

AnI-E group is a group in which the endomorphism near-ring generated by the group's inner automorphisms equals the endomorphism near-ring generated by its endomorphisms. In this paper we shall completely determine the finite groups that are semidirect products of cyclic groups and areI-E groups.

Semidirect products of 𝐼-𝐸 groups

An I-E group is a group G in which the endomorphism near-ring generated by the inner automorphisms of G equals the endomorphism near-ring generated by the endomorphisms of G. In this paper we obtain a result characterizing when a semidirect product of I-E groups of relatively prime orders is an I-E group. We then use this result to show that a semidirect product of cyclic groups of relatively prime orders is an I-E group.

On the structure of an endomorphism near-ring

If G is an additive (but not necessarily abelian) group and S is a semigroup of endomorphisms of G, the endomorphism near-ring R of G generated by S consists of all the expressions of the form ɛ1s1+…+ɛnsnwhere ɛi=±1 and si∈S for each i. When functions are written on the right, R forms a distributively generated left near-ring under pointwise addition and composition of functions. A basic reference on near-rings which has a substantial treatment of endomorphism near-rings is [6].

Automorphism Groups Emitting Local Endomorphism Near-Rings

The question of whether a group of automorphisms $A$ containing the inner automorphisms of a nonabelian $p$-group $G$ can generate a local endomorphism near-ring when $A$ is not a $p$-group is considered. Conditions on $A$ are obtained which give us that the endomorphism near-ring generated by $A$ is not local. An example is given showing that the endomorphism near-ring generated by $A$ can be local when these conditions are not met.

Local endomorphism near-rings

The purpose of this paper is to study the consequences of an endomorphism near-ring of a finite group being a local near-ring and the existence of such near-rings. As we shall see in Section 2, an endomorphism near-ring of a finite group being local gives us some information about both the structure of the group (Theorem 2.2) and the automorphisms of the group lying in the near-ring (Theorem 2.3). Existence of local endomorphism near-rings of finite groups is considered in Section 3 where we obtain as our main result that any p-group of automorphisms of a p-group containing the inner automorphisms always generates a local endomorphism near-ring. In particular, we get as a corollary that the endomorphism near-ring of a finite group G generated by the inner automorphisms of G is local if and only if G is a p-group. The third section concludes with a discussion of endomorphism near-rings of dihedral 2-groups and generalized quaternion groups.

Near-rings of mappings on finite topological groups

AbstractWhen G is a topological group, the set N(G) of continuous self-maps of G, and the subset N0(G) of those which fix the identity of G, are near-rings. In this paper we examine the (left) ideal structure of these near-rings when G is finite. N0(G) is shown to have exactly two maximal ideals, whose intersection is the radical. In the final section we investigate subnear-rings of N0(G) determined by certain continuous elements of the endomorphism near-ring.

On the endomorphism near-ring of a free group

Suppose F is an additively written free group of countably infinite rank with basis T and let E = End(F). If we add endomorphisms pointwise on T and multiply them by map composition, E becomes a near-ring. In her paper “On Varieties of Groups and their Associated Near Rings” Hanna Neumann studied the sub-near-ring of E consisting of the endomorphisms of F of finite support, that is, those endomorphisms taking almost all of the elements of T to zero. She called this near-ring Φω. Now it happens that the ideals of Φω are in one to one correspondence with varieties of groups. Moreover this correspondence is a monoid isomorphism where the ideals of φω are multiplied pointwise. The aim of Neumann's paper was to use this isomorphism to show that any variety can be written uniquely as a finite product of primes, and it was in this near-ring theoretic context that this problem was first raised. She succeeded in showing that the left cancellation law holds for varieties (namely, U(V) = U′(V) implies U = U′) and that any variety can be written as a finite product of primes. The other cancellation law proved intractable. Later, unique prime factorization of varieties was proved by Neumann, Neumann and Neumann, in (7). A concise proof using these same wreath product techniques was also given in H. Neumann's book (6). These proofs, however, bear no relation to the original near-ring theoretic statement of the problem.

The endomorphism near-ring of an infinite dihedral group

SynopsisIn this paper, we determine the semigroup End D of endomorphisms of the infinite dihedral group D, and give a multiplication table for it. We determine the additive structure of the near-rings E(D) generated by the endomorphisms of D, A(D) generated by the automorphisms of D and I(D) generated by the inner automorphisms of D, and determine their radicals and all their maximal right ideals.

Near-rings of quotients of endomorphism near-rings

Let be a category with finite products and a final object and let X be any group object in . The set of -morphisms, (X, X) is, in a natural way, a near-ring which we call the endomorphism near-ring of X in Such nearrings have previously been studied in the case where is the category of pointed sets and mappings, (6). Generally speaking, if Γ is an additive group and S is a semigroup of endomorphisms of Γ then a near-ring can be generated naturally by taking all zero preserving mappings of Γ into itself which commute with S (see 1). This type of near-ring is again an endomorphism near-ring, only the category is the category of S-acts and S-morphisms (see (4) for definition of S-act, etc.).

Endomorphism near-rings

The study of near-rings is motivated by consideration of the system generated by the endomorphisms of a (not necessarily commutative) group. Such endomorphism near-rings also furnish the motivation for the concept of a distributively generated (d.g.) near-ring. Although d.g. near-rings have been extensively studied, little is known about the structure of endomorphism near-rings. In this paper results are presented which enable one to give the elements of the endomorphism near-ring of a given group. Also, some results relating to the right ideal structure of an endomorphism near-ring are presented. These concepts are applied to present a detailed picture of the properties of the endomorphism near-ring of (S3, +).

Endomorphism near-ring of a relatively free group

Endomorphism near-ring of a relatively free group