The Levi class L ( M ) generated by the class of groups M is the class of all groups in which the normal closure of each cyclic subgroup belongs …
The Levi class L ( M ) generated by the class of groups M is the class of all groups in which the normal closure of each cyclic subgroup belongs to M . Let p be a prime number, p ̸ = 2 , s be a natural number, s ≥ 2 , and s > 2 for p = 3; H p s be a free group of rank 2 in the variety of nilpotent groups of class ≤ 2 of exponent p s with commutator subgroup of exponent p ; Z is an in fi nite cyclic group; q { H p s , Z } is a quasivariety generated by the set of groups { H p s , Z } . We fi nd a basis of quasi-identities of the Levi class L ( q { H p s , Z } ) and establish that there exists a continuous set of quasivarieties K such that L ( K ) = L ( q { H p s , Z } ) .
The following conditions for a group G are investigated: (i) maximal class n subgroups are normal, (ii) normal closures of elements have nilpotency class n at most, (iii) normal closures …
The following conditions for a group G are investigated: (i) maximal class n subgroups are normal, (ii) normal closures of elements have nilpotency class n at most, (iii) normal closures are n –Engel groups, (iv) G is an ( n +1 )-Engel group. Each of these conditions is a consequence of the preceding one. The second author has shown previously that all conditions are equivalent for n = 1. Here the question is settled for n = 2 as follows: conditions (ii), (iii) and (iv) are equivalent. The class of groups defined by (i) is not closed under homomorphisms, and hence (i) does not follow from the other conditions.
We investigate the concept of dominion (in the sense of Isbell) in several varieties of nilpotent groups. We obtain a complete description of dominions in the variety of nilpotent groups …
We investigate the concept of dominion (in the sense of Isbell) in several varieties of nilpotent groups. We obtain a complete description of dominions in the variety of nilpotent groups of class at most two. Then we look at the behavior of dominions of subgroups of groups in N 2 when taken in the context of Nc for c>2. Finally, we establish the existence of nontrivial dominions in the category of all nilpotent groups.
IntroductionSuppose that A and B are subgroups of a group G.If there exists a positive integer m such that the commutator (...((a, b), ...), b) 1 for all a in …
IntroductionSuppose that A and B are subgroups of a group G.If there exists a positive integer m such that the commutator (...((a, b), ...), b) 1 for all a in A and b in B, then we write A le:mJ B. A group G which satisfies G le:ml G is said to satisfy the m t Engel condition.The problem of determining for what groups the mth Engel condition implies nilpotence has been studied by several authors.For example, K. Gruenberg in [2] has shown that finitely generated soluble groups which satisfy the mth Engel condition are nilpotent.R. Baer in [1] adds groups which satisfy the maximal condition to the list.I1 [3] Gruenberg includes the torsion-free soluble groups This paper grew out of an investigation of the commutator structure of Z-A groups, that is groups in which G itself is a term of its upper central series.The class of a Z-A group is the smallest ordinal n such that Z G where Zn denotes the n th term in the upper central series of G.The investi- gation resulted in a curious classification of Z-A groups.This classification is based on a class of Z-A groups which it seemed natural to call Z-A(q) groups for integer q.We will show that Z-A(1) is equal to the above class of groups and that Z-A( 1 > Z-A(2) > Z-A(3).The class of Z-A(3) groups proved to be interesting.For instance, an example of a metabelian.Z-A(3) group is found which has exponent 4 and satisfies the 3 rd Engel condition, but is not nilpotent.However, every Z-A(3) group with pime exponent is auto- matically nilpotent.It may not be significant but no example of a Z-A(3) group has been found which is not of class -t-1 and where Z is not abelian.The following pages will investigate under what conditions the Engel condition implies nilpotence for Z-A (3) groups.We will recall some definitions and notations.If x and y are elements of a group, then denote the product x-l.y-l.x.y of a group by the commutator (x, y).We define commutators of higher order by the recursive rule (xl, ..., xn_l, xn) ((x, ..., x_), xn).Define the weight w(c)of the commutator c constructed from the elements x, x. recursively by defining the weight of the elements x,x. to be 1, and if c (c, c.)
This is a collection of open problems in Group Theory proposed by more than 300 mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk …
This is a collection of open problems in Group Theory proposed by more than 300 mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965, now also in English. This is the 18th edition, which contains 120 new problems and a number of comments on about 1000 problems from the previous editions.
Denote the commutator of two group elements a and b by aba -1 b -1 =(a,b) (1) then the associative law for the commutator operation is (a, b), c) = …
Denote the commutator of two group elements a and b by aba -1 b -1 =(a,b) (1) then the associative law for the commutator operation is (a, b), c) = (a, (b,c)). (2) If (2) is satisfied for every triplet of elements of a group, this group will be called an S group. If (2) is supposed to be satisfied whenever two of the elements a, b, c are equal (alternating law), then the group will be called an L-group.