Rapid climatic and socio-economic changes challenge current agricultural R&D capacity. The necessary quantum leap in knowledge generation should build on the innovation capacity of farmers themselves. A novel citizen science methodology, triadic comparisons of technologies or tricot, was implemented in pilot studies in India, East Africa, and Central America. The methodology involves distributing a pool of agricultural technologies in different combinations of three to individual farmers who observe these technologies under farm conditions and compare their performance. Since the combinations of three technologies overlap, statistical methods can piece together the overall performance ranking of the complete pool of technologies. The tricot approach affords wide scaling, as the distribution of trial packages and instruction sessions is relatively easy to execute, farmers do not need to be organized in collaborative groups, and feedback is easy to collect, even by phone. The tricot approach provides interpretable, meaningful results and was widely accepted by farmers. The methodology underwent improvement in data input formats. A number of methodological issues remain: integrating environmental analysis, capturing gender-specific differences, stimulating farmers' motivation, and supporting implementation with an integrated digital platform. Future studies should apply the tricot approach to a wider range of technologies, quantify its potential contribution to climate adaptation, and embed the approach in appropriate institutions and business models, empowering participants and democratizing science.

Swift heavy ion (SHI) irradiation effects on ionic conduction in the PVA-H3PO4 polymer electrolyte films have been investigated due to its variety of applications in electrochemical devices. Polymer electrolytes films are irradiated with 50 MeV Li3+ ions having five different fluences viz. 5x1010, 1011, 5x1011, 1012 and 5x1012 ions/cm2. It is observed that irradiation of the polymer electrolyte films with swift heavy ions shows enhancement in conductivity at lower fluences and decrease in conductivity at higher fluences. It appears that below the critical fluence, swift heavy ion irradiation increases the diffusivity of Li+ ion in the polymer electrolyte which provides larger pathways for ionic transport throughout the system. The temperature dependence of electrical conductivity variation has been used to compute the activation energy involved in conduction process.

For a periodic group G, denote by ω(G) the set of orders of elements in G. We prove that if ω(G) is a proper subset of the set {1, 2, 3, 4, 5} then either G is locally finite or G contains a nilpotent normal subgroup N such that G/N is a 5-group.

Nielsen [14] gave the following commutator test for an endomorphism of the free group F = F2 = 〈x, y; Ø〉 to be an automorphism: an endomorphism ø: F → F is an automorphism if and only if the commutator [ø(x), ø(y)] is conjugate in F to [x, y]±1. He obtained this test as a corollary to his well-known result that every IA-automorphism of F (i.e. one which fixes F modulo its commutator subgroup) is an inner automorphism. Bachmuth et al. [4] have proved that IA-automorphisms of most two-generator groups of the type F/R′ are inner, and it becomes natural to ask if Nielsen's commutator test remains valid for those groups as well. Durnev[7] considered this question for the free metabelian group F/F″ and confirmed the validity of the commutator test in this case. Here we prove that Nielsen's test does not hold for a large class of F/R′ groups (Theorem 3·1) and, as a corollary, deduce that it does not hold for any non-metabelian solvable group of the form F/R″ (Corollary 3·2). In view of our Theorem 3·1, Nielsen's commutator test in these situations seems to have less appeal than his result that the IA-automorphisms of F are precisely the inner automorphisms of F. We explore some applications of this important result with respect to non-tameness of automorphisms of certain two- generator groups F/R (i.e. automorphisms of F/R which are not induced by those of the free group F). For instance, we show that a two-generator free polynilpotent group F/V, , has non-tame automorphisms except when V = γ2(F) or V = γ3(F), or when V is of the form [yn(U), γ(U)], n ≥ 2 (Theorem 4·2). This complements the results of [9] and [16] rather nicely, and is shown to follow from a more general result (Proposition 4·1). We also include an example of an endomorphism θ: x → xu, y→y of F which induces a non-tame automorphism of F/γ6(F) while the partial derivative ∂(u)/∂(x) is ‘balanced’in the sense of Bryant et al. [5] (Example 4·4). This gives an alternative solution of a problem in [5] which has already been resolved by Papistas [15] in the negative. In our final section, we consider groups of the type F/[R′,F] and, in contrast to groups of the type F/R′, we show that the Nielsen's commutator test does hold in most of these groups (Theorem 5·1). We conclude with a sufficiency condition under which Nielsen's commutator test is valid for a given pair of generating elements ofF modulo [R′,F] (Proposition 5·2).

It is a well-known result due to Sjogren [9] that if G is a finitely generated p-group then, for all n ≦ − 1, the (n + 2)-th dimension sub-group Dn+2(G) of G coincides with γn+2(G), the (n + 2)-th term of the lower central series of G. This was earlier proved by Moran [5] for n ≦ p − 2. For p = 2, Sjogren’s result is the best possible as Rips [8] has exhibited a finite 2-group G for which D4G) ≠ γ4(G) (see also Tahara [10, 11]). In this note we prove that if G is a finitely generated metabelian p-group then, for all . It follows, in particular, that, for p odd, for all n ≦ p and all metabelian p-groups G.

The purpose of this expository article is to familiarize the reader with one of the fundamental problems in the theory of infinite groups. We give an up-to-date account of the so-called Fox problem which concerns the identification of certain normal subgroups of free groups arising out of certain ideals in the free group rings. We assume that the reader is familiar with the elementary concepts of algebra.

We prove that a cyclic extension of a free product of finite p-groups is residually finite, p a fixed prime. The question of cyclic extensions of free products of arbitrary cyclic groups remains open.

Any variety of groups is generated by its free group of countably infinite rank. A problem that appears in various forms in Hanna Neumann's book [7] (see, for intance, sections 2.4, 2.5, 3.5, 3.6) is that of determining if a given variety B can be generated by Fk(B), one of its free groups of finite rank; and if so, if Fn(B) is residually a k-generator group for all n ≧ k. (Here, as in the sequel, all unexplained notation follows [7].)

The largest group of exponent 4 generated by n elements of order 2 has nilpotency class n + 1 when n ≥ 3.

If is a variety of groups that can be denned by n-variable laws and (m) is the variety all of whose m-generator groups are in then there corresponds the chain: (1) ≧ (2) ≧ . . . ≧ (n) = . In this paper such chains are investigated to determine which of the inclusions are proper for certain varieties . In particular the inclusions are shown to be all proper for the varieties where is the variety of nilpotent-of-class-c groups, is the abelian variety and is the variety of centre-bymetabelian groups. For the inclusions are likewise proper but for the corresponding chain is:

It is shown that a 3rd-Engel group is an extension of a soluble group by a group of exponent 5.

Let denote the variety of all groups which are extensions of a nilpotent-of-class-c group by a nilpotent-of-class-d group, and let denote the variety of all metabelian groups. The main result of this paper is the following theorem.

THEOREM. Let be a subvariety of which does not contain . Then every -group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent. In particular, a finitely generated torsion-free -group is a nilpotent-by-finite group.

This generalizes the main theorem of Ŝmel′kin [4], where the same result is proved for subvarieties of , where is the variety of abelian groups. See also Lewin and Lewin [2] for a related discussion.

Let (n, σ, d) denote the variety of all groups defined by the left-normed commutator identity [x1, …, xn] = [x1σ, …, xnσ]d, where σ is a non-identity permutation of {1, …, n}, and d is an integer, possibly negative. It is shown that (n, σ, d) is nilpotent-by-nilpotent if σ ≠ (1, 2), abelian by nilpotent if n > 2, nσ ≠ n, and nilpotent of class at most n + 1 if {1, 2} ≠ {1σ, 2σ}. This improves on a result of E.B. Kikodze that (n, σ, 1) is locally soluble and if {1, 2} ≠ {1σ, 2σ} is locally nilpotent.

We refer the reader to Hanna Neumann [7] for notation and other undefined terms. Let and denote the varieties of groups defined by the laws (xy)n=xnyn, [x, y]n=1 and [x, yn]respectively, where n is an integer. (n)-groups were termed “n-abelian” by R. Baer [1] and have been a subject matter of investigation by various authors (see [3], [5], [6] and the references therein). Recently KaluŽnin [5] has shown that , thus clarifying the relationship between U(n) and the familiar varieties. From the elementary inequalities(n ≠ 0,1)it is easily deduced that(see for instance [5]). If G = Cm Wr C∞, then cleary but for any Thus and . It is also easy to see that in general (see for instance [6] § 5.1) and we are led to ask