Paper Information

Title: 

GROUPS WHOSE ELEMENTS COMMUTE WITH THEIR ENDOMORPHIC IMAGES

Type: PAPER
Author(s): FAGHIHI AFARANI A.*
 
 *FACULTY OF SCIENCES, UNIVERSITY OF QOM, ISFAHAN ROAD, QOM, IRAN, ZIP CODE: 3716146611
 
Name of Seminar: SEMINAR ON ALGEBRA (SALG20)
Type of Seminar:  CONFERENCE
Sponsor:  TARBIAT MOALLEM UNIVERSITY, FACULTY OF MATHEMATICAL SCIENCES AND COMPUTER
Date:  2009Volume 20
 
 
Abstract: 

A group G is called an E-group if the Near-ring generated by the endomorphisms of G in the near-ring of maps on G is a ring. It is well known (see, e.g., Malone, 1995) that a group G is an E-group if and only if each element commutes with its endomorphic images. For any prime number p, we call an E-group which is also a p-group, a pE-group. In this paper at first we explain general properties of E-groups. Also we prove that an infinite finitely generated E-group is the direct product of a central torsion-free subgroup and a finite subgroup. Next, we prove that there is no 3E-group of nilpotency class 3 of order at most 310. Also we construct a group of class 3 which is “very close” to be an E-group. The following questions are central ones in this paper:
(1) what is the least number of generators of a finitely generated nonabelian E-group?
(2) What is the minimum order of a finite non-abelian pE-group?
We prove that the minimal number of generators of a finitely generated non-abelian E-group is 4.
In response to the question (2), we prove that the minimum order of a finite non-abelian pE-group is p8, for any odd prime number p and this order is 27 for p=2.
Also we obtain a new class of E-groups.
As we have found that some of our results are valid for a very larger class of finite p-groups than pE-groups, we study a class of p-groups for every prime number p and we denote this class of p-groups by p
e. (A finite p-group G is called a pe-group if G is a 2-Engel group and all elements of order at most pr lie in the center of G, where pr is exponent G/G1).
We classify all 3-generator pE-groups and pE-groups with cyclic derived subgroup and determine endomorphisms of 3-generator pE-groups and pE-groups. Finally we classify all pE-groups and p
e-groups of order at most p7 for any prime number p.

 
Keyword(s): ENDOMORPHISM OF GROUPS, 2-ENGEL GROUPS, P-GROUPS, NEARRINGS
 
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