The endomorphism near-rings of the symmetric groups of degree at least five

Abstract

The near-ring distributively generated by the semigroup of all endomorphisms of S n , the symmetric group of degree n, for n > 5, is close to being the near-ring of all mappings from S n to itself respecting the identity. In this paper, the structure of these near-rings is studied in detail. In particular, addition and multiplication rules for the elements given in canonical form are determined. A complete list of all right ideals, left ideals, right invariant and left invariant subgroups is given. 1980 Mathematics subject classification (Amer. Math. Soc.): 16 A 76.

Inverse Semigroups and Near-Rings

“…Let e,f be two idempotent endomorphisms of a group G. Then Proof. This result is a slight elaboration of Theorem 9.7 of Fong [2], and can be found in Fong [3]. We first prove (i).…”

Polynomial Automorphisms of Soluble Groups

“…For instance, it is well known that every automorphism of the symmetric group S n of degree n is inner, except for n = 6, where in this case the group of inner automorphisms is of index 2 in Aut S 6 (see for instance [19,Theorem 7.10]). However, all non-inner automorphisms of S 6 are polynomial ( [7]; also see Example 3 in [14]) and so we can state that each automorphism of S n is polynomial for any positive integer n. Let us give another example: by a famous result of Magnus, two elements a and b of a free group have the same normal POLYNOMIAL AUTOMORPHISMS 3389 closure if and only if there is an inner automorphism mapping a into b or b −1 (see for example [16, Chapter II, Proposition 5.8]). It turns out that this property remains true in any torsion-free nilpotent metabelian group provided that the term "inner automorphism" is replaced by the term "polynomial automorphism" [6].…”

Near-rings of mappings on finite topological groups

“…There is an extensive literature on these near-rings for various classes of finite groups (see for example [1], [4], [5], [6], [7]) and using these results we will examine two kinds of subnear-rings of N H . The first is E H = N H n E, the near-ring of continuous maps in E, and the second is C H , the near-ring distributively generated by the continuous elements in End G. (E H , although a subnearring of E, is not necessarily distributively generated.)…”

“…In the third section we look at two subnear-rings of N 0 (G) determined by [2] Near-rings of mappings 93 endomorphisms, namely the intersection of N 0 (G) with the endomorphism nearring E(G), and the near-ring distributively generated by continuous elements of E(G). In particular, the orders of these near-rings are obtained for classes of near-rings for which the order of E(G) is known (see [1], [5], [6], [7]). …”