1·1. The main result. Let ø = ø(x1, x2, …, xn) be a word in n variables and let G be a group. A ø-element of G is any element of the form ø(x1, x2, …, xn)±1 with xi ∈G(1 ≤ i ≤ n). The subgroup generated by all the ø-elements of G is the verbal subgroup ø(G) of G. If there is a positive integer l such that every element of ø(G) can be expressed as the product of l or fewer ø-elements of G we say that G is ø-elliptic. If there is no such integer we say that G is ø-parabolic.

A subset S of an additive group G is called a maximal sum-free set in G if {S+S) n S = 0 and \s\ > |2"| for every sum-free set T in G . It is shown that if G is an elementary abelian p-group of order p , where p = 3k ± 1 , then a maximal sum-free set in G has kp elements. The maximal sum-free sets in Z are characterized to within automorphism.Given an additive group G and non-empty subsets S, 1 of G , let S + T denote the set {s+t; s € 5 , t f ?} , S the complement of S in G and \s\ the cardinality of 5 . We call S a sum-free set in C if (5+5) £ S . If, in addition, |s| i |r| for every sum-free set T in G , then we call S a maximal sum-free set in G . We denote by \(G) the cardinality of a maximal sum-free set in G .If G is a finite abelian group, then according to [2], 2|G|/7 2 \{G) 5 \G\/2 . Both these bounds can be attained since X(Z 7 ) = 2 , A(Z 2 ) = 1 , where Z denotes the cyclic group of order n .

Abstract. A group G is called strongly restrained if there exists an integer n suchthat {xW) can be generated by n elements for all x,y in G. We show that a group G is polycyclic-by-finite if and only if G is a finitely generated strongly restrained group in which every nontrivial finitely generated subgroup has a nontrivial finite quotient. This provides a general setting for various results in soluble and residually finite groups that have appeared recently.

Abstract. Groups which are locally indicable are also right-orderable, but not conversely. This paper considers a characterization of local indicability in right-ordered groups, the key concept being a property of right-ordered groups due to Conrad. Our methods answer a question regarding the Artin braid groups Bn which are known to be right-orderable. The subgroups Pn of pure braids enjoy an ordering which is invariant under multiplication on both sides, and it has been asked whether such an ordering of Pn could extend to a rightinvariant ordering of Bn. We answer this in the negative. We also give another proof of a recent result of Linnell that for elementary amenable groups, the concepts of right-orderability and local indicability coincide. Definitions and statement of resultsA right-ordered group is a pair (G, <), where G is a group, < is a strict total ordering of the elements of G, and right-invariance holds:If the ordering is also left-invariant,then we call (G, <) a bi-ordered group (also known in the literature as "totally ordered" or, simply, "ordered").A group G is said to be locally indicable if for every nontrivial finitely-generated subgroup H of G there is a nontrivial homomorphism H → Z onto the additive group of integers. Proposition 1.1. Locally indicable groups are right-orderable.Proof. Burns and Hale in [4] show that a group G is right-orderable if and only if every finitely generated nontrivial subgroup of G has a nontrivial quotient that is also right-orderable. See [16], Theorem 7.3.1,or [13], Theorem 3.2.1. Since Z is right-orderable, the proposition follows.The converse is not true. Bergman [2] gave an example of a finitely-generated right-orderable group which is perfect-its abelianization is trivial. Any homomorphism of such a group to Z must be trivial, so it is not locally indicable.

Abstract.We look at groups which have no (nonabelian) free subsemigroups. It is known that a finitely generated solvable group with no free subsemigroup is nilpotent-by-finite. Conversely nilpotent-by-finite groups have no free subsemigroups. Torsion-free residually finite-p groups with no free subsemigroups can have very complicated structure, but with some extra condition on the subsemigroups of such a group one obtains satisfactory results. These results are applied to ordered groups, right-ordered groups, and locally indicable groups.