Akbar Rhemtulla

Local indicability in ordered groups: Braids and elementary amenable groups

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 B n which are known to be right-orderable. The subgroups P n of pure braids enjoy an ordering which is invariant under multiplication on both sides, and it has been asked whether such an ordering of P n could extend to a right-invariant ordering of R n . 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.

Maximal sum-free sets in finite abelian groups

A subset S of an additive group G is called a maximal sum-free set in G if ( S + S ) ∩ S = o and ∣ S ∣ ≥ ∣ T ∣ for every sum-free set T in G . It is shown that if G is an elementary abelian p –group of order p n , where p = 3k ± 1, then a maximal sum-free set in G has kp n-1 elements. The maximal sum-free sets in Z p are characterized to within automorphism.

GROUPS WITH MANY PERMUTABLE SUBGROUPS

We consider the influence on a group G of the condition that every infinite set of cyclic subgroups of G contains a pair that permute and prove (Theorem 1) that finitely generated soluble groups with this condition are centre-by-finite, and (Theorem 2) that torsion free groups satisfying the condition are abelian.

When is a right orderable group locally indicable

If a group G has an ascending series 1 = G0 ≤ G1 ≤ · · · ≤ Gρ = G of subgroups such that for each ordinal α, Gα C G, and Gα+1/Gα has no nonabelian free subsemigroup, then G is right orderable if and only if it is locally indicable. In particular if G is a radical-by-periodic group, then it is right orderable if and only if it is locally indicable.

CYCLICALLY SEPARATED GROUPS

We call a group G cyclically separated if for any given cyclic subgroup B in G and subgroup A of finite index in B , there exists a normal subgroup N of G of finite index such that N ∩ B = A . This is equivalent to saying that for each element x ∈ G and integer n ≥ 1 dividing the order o ( x ) of x , there exists a normal subgroup N of G of finite index such that Nx has order n in G/N . As usual, if x has infinite order then all integers n ≥ 1 are considered to divide o ( x ). Cyclically separated groups, which are termed “potent groups” by some authors, form a natural subclass of residually finite groups and finite cyclically separated groups also form an interesting class whose structure we are able to describe reasonably well. Construction of finite soluble cyclically separated groups is given explicitly. In the discussion of infinite soluble cyclically separated groups we meet the interesting class of Fitting isolated groups, which is considered in some detail. A soluble group G of finite rank is Fitting isolated if, whenever H = K/L ( L ⊲ K ≤ G ) is a torsion-free section of G and F ( H ) is the Fitting subgroup of H then H / F ( H ) is torsion-free abelian. Every torsion-free soluble group of finite rank contains a Fitting isolated subgroup of finite index.

The structure of Bell groups

Abstract For any integer n > 1, the variety of n-Bell groups is defined by the law w(x 1, x 2) = [x 1 n , x 2][x 1, x 2 n ]−1. Bell groups were studied by R. Brandl in [2], and by R. Brandl and L.-C. Kappe in [3]. In this paper we determine the structure of these groups. We prove that if G is an n-Bell group then G/Z 2(G) has finite exponent depending only on n. Moreover, either G/Z 2(G) is locally finite or G has a finitely generated subgroup H such that H/Z(H) is an infinite group of finite exponent. Finally, if G is finitely generated, then the subgroup H may be chosen to be the finite residual of G.

Criteria for commutativity in large groups

In this paper we prove the following: 1. Let m 1⁄2 2, n 1⁄2 1 be integers and let G be a group such that (XY)n ≥ (YX)n for all subsets X, Y of size m in G. Then a) G is abelian or a BFC-group of finite exponent bounded by a function of m and n. b) If m 1⁄2 n then G is abelian or jGj is bounded by a function of m and n. 2. The only non-abelian group G such that (XY)2 ≥ (YX)2 for all subsets X, Y of size 2 in G is the quaternion group of order 8. 3. Let m, n be positive integers and G a group such that X1 Ð Ð ÐXn [ o2Snn1 Xo(1) Ð Ð ÐXo(n) for all subsets Xi of size m in G. Then G is n-permutable or jGj is bounded by a function of m and n. The first author wishes to thank the Department of Mathematics, University of Alberta for their excellent hospitality. He also thanks the University of Isfahan for its financial support. Received by the editors June 20, 1996. AMS subject classification: Primary: 20E34; secondary: 20F24. c Canadian Mathematical Society 1998.

Groups with many rewritable products

For any integer n > 2, denote by Rn the class of groups G in which every infinite subset X contains n elements x1, ..., xn such that the product x1 ... = Xa(l) ... xa(n) for some permutation a 54 1. The case n = 2 was studied by B. H. Neumann who proved that R2 is precisely the class of centre-by-finite groups. Here we show that G E Rn for some n if and only if G contains an FC-subgroup F of finite index such that the exponent of F/Z(F) is finite, where Z(F) denotes the centre of F.

PERMUTABLE WORD PRODUCTS IN GROUPS

wherer > 0 is fixed and U = {«}, V = {v} then G £ P{U,V) G/Z(G) if an ids onl of y ifexponent r. This is not difficult to verify.In general the classes P(U, V) may be viewed as generalising varieties and, exceptfor some specific sets U and V, it is very difficult to describe them. We now turn torelated classes of groups.Let n > 0 be fixed, X —

MORE FIBONACCI VARIETIES

The Fibonacci varieties introduced by Ann Chi Kim, Bull. Austral. Math. Soc. 19 (1978), 191-196 (1979), are here generalised in an obvious way suggested by the work of D.L. Johnson, J.W. Wamsley, and D. Wright, Proa. London Math. Soc. (3) 29 (197*0, 577-592. The underlying groups of algebras in these varieties are studied, and are shown to be abelian when the algebras are generated by a single element, and in general are found to be extensions of central subgroups by groups of finite exponent.