On non-Hopfian groups of fractions

The group of fractions of a semigroup S, if exists, can be written as G = SS−1. If S is abelian, then G must be abelian. We say that a semigroup identity is transferable if being satisfied in S it must be satisfied in G = SS−1. One of problems posed by G.Bergman in 1981 asks whether the group G must satisfy every semigroup identity which is satisfied in S, that is whether every semigroup identity is transferable. The first non-transferable identities were constructed in 2005 by S.V.Ivanov and A.M. Storozhev. A group G is called Hopfian if each epimorphizm G → G is the automorphism. The residually finite groups are Hopfian, however there are many problems concerning the Hopfian property e.g. of infinite Burnside groups, of finitely generated relatively free groups [11, Problem 15]. We prove here that if G = SS−1 is an n-generator group of fractions of a relatively free semigroup S, satisfying m-variable (m < n) non-transferable identity, then G is the non-Hopfian group.

Production de composés indoliques rhizogènes par le ver de terre Lumbricus terrestris

In vitro application of total gross extract of earthworms (Lumbricus terrestris) in diverse dilutions stimulates rhizogenesis in young bean plants (Phaseolus vulgaris). The observed effect is similar to that of indol acetic acid, a well-known growth enhancer in plants, used here as a control in various concentrations. Fragmentation of worm extract by column chromatography results in three groups of fractions. Only the polar group of fractions has a significant rhizogenous effect, which is, however, inferior to that observed in the presence of total gross extract of worms or of indol acetic acid. Gross extract analyses using thin layer chromatography, with appropriate chromatography systems and reagents, revealed that indol acetic acid is not present, but is probably replaced by other indol-derived substances that have a neutral to basic chromatographic behaviour. These presumed indol-derived substances are identified as methyl-tryptophane, serotonin, and hydroxy-indol acetic acid. Analyses using mass spectrometry combined with gas chromatography, following fragmentation and purification of the group of rhyzogenous fractions, have revealed the presence of hydroxy-indol carboxylic acid, which seems to take the form of several isomeres.[Journal translation]

Growth of linear semigroups

We show that the growth function of a finitely generated linear semigroup S ⊆ Mn(K) is controlled by its behaviour on finitely many cancellative subsemigroups of S. If the growth of S is polynomially bounded, then every cancellative subsemigroup T of S has a group of fractions G ⊆ Mn (K) which is nilpotent-by-finite and of finite rank. We prove that the latter condition, strengthened by the hypothesis that every such G has a finite unipotent radical, is sufficient for S to have a polynomial growth. Moreover, the degree of growth of S is then bounded by a polynomial f(n, r) in n and the maximal degree r of growth of finitely generated cancellative T ⊆ S.