SYSTEMS OF EQUATIONS IN A FREE GROUP. II

1972 ◽  
Vol 6 (1) ◽  
pp. 109-180 ◽  
Author(s):  
Ju I Hmelevskiĭ
2020 ◽  
Vol 102 (2) ◽  
pp. 796-817
Author(s):  
Chloé Perin ◽  
Rizos Sklinos

2001 ◽  
Vol 11 (03) ◽  
pp. 375-390
Author(s):  
AMNON ROSENMANN

Let h1, h2,… be a sequence of elements in a free group and let H be the subgroup they generate. Let H′ be the subgroup generated by w1, w2, …, where each wi is a word in hi and possibly other hj, such that the associated directed graph has the finite paths property. We show that rank H′≥ rank H. As a corollary we get that [Formula: see text], where [Formula: see text] is the subgroup generated by the roots of the elements in H. If H0 is finitely generated and the sequence of subgroups H0, H1, H2, … satisfies [Formula: see text] then the sequence stabilizes, i.e. for some m, Hi=Hi+1 for every i≥ m. When applied to systems of equations in free groups, we give conditions on a transformation of the system such that the maximal rank of a solution (the inner rank) does not increase. In particular, we show that if in "Lyndon equation" [Formula: see text] the exponents ai satisfy gcd (a1,…,an)≠1 then the inner rank is ⌊ n/2⌋. The proofs are mostly elementary.


1970 ◽  
Vol 22 (5) ◽  
pp. 1040-1046 ◽  
Author(s):  
I. M. Isaacs

Let F be the free group on n generators x1, …, Xn and let G be an arbitrary group. An element ω ∈ F determines a function x → ω(x) from n-tuples x = (x1, x2, …, xn) ∈ Gn into G. In a recent paper [5] Solomon showed that if ω1, ω2, …, ωm ∈ F with m < n, and K1, …, Km are conjugacy classes of a finite group G, then the number of x ∈ Gn with ωi(x) ∈ Ki for each i, is divisible by |G|. Solomon proved this by constructing a suitable equivalence relation on Gn.Another recent application of an unusual equivalence relation in group theory is in Brauer's paper [1], where he gives an elementary proof of the Frobenius theorem on solutions of xk = 1 in a group.


2007 ◽  
Vol 19 (1) ◽  
pp. 235 ◽  
Author(s):  
J. Mestach ◽  
J. Paeshuyse ◽  
J. Neyts ◽  
H. J. Nauwynck ◽  
D. Maes ◽  
...  

Bovine viral diarrhea virus (BVDV) causes serious economic losses in the cattle industry. Evidence exists that only zona pellucida (ZP)-free bovine embryos are susceptible to BVDV infection (Vanroose et al. 1998 Biol. Reprod. 58, 857–866); however, BVDV can adhere to and therefore ‘infect’ both in vivo-(Waldrop et al. 2004 Theriogenology 62, 387–397) and in vitro-produced ZP intact embryos (Stringfellow et al. 2000 Theriogenology 53, 827–839). To eliminate these sanitary risks, pre-treatment of embryos with antiviral compounds may be a promising approach (Givens et al. 2006 Theriogenology 65, 344–355). BPIP (5-[(4-bromophenyl)methyl]-2-phenyl-5H-imidazo[4,5-c]pyridine) has been reported to display antiviral activity against BVDV, with a 50% effective inhibition of BVDV-induced cytopathic effect formation at a concentration of 0.04 µM (Paeshuyse et al. 2006 J. Virol. 80, 149–160). However, since the short- and long-term effects of BPIP have not been described, the aim of the current study was to assess whether addition of BPIP for 2 days at a concentration of 5 µM is toxic for ZP-free cattle embryos. Oocytes were aspirated from 3–6-mm follicles of cattle ovaries, matured for 24 h, and subsequently co-incubated with 1 × 106 sperm cells mL−1 in IVF-TALP with 20 µg/mL−1 heparin for 24 h at 39°C and 5% CO2 in air. After fertilization, presumptive zygotes were put in groups of 25 into 50-µL droplets of SOF under oil in 5% CO2, 5% O2, and 90% N2 for 6 days. Afterwards, morulae and blastocysts were collected, rendered ZP-free by means of pronase treatment, and divided into 4 groups: (i) ZP-free control group, (ii) ZP-free control group treated with a volume of DMSO equal to condition (iv), (iii) ZP-free group treated with 5 µM BPIP in DMSO, and (iv) ZP-free group treated with 10 µM BPIP in DMSO. Because BPIP is a fat-soluble molecule, embryos were cultured in 0.5 mL SOF without oil for 2 days. At Day 8, all embryos were fixed, TUNEL-stained, and analyzed for total cell number and percentage of apoptotic cells. Three independent replicates were performed. Results are shown in Table 1 and were analyzed by means of ANOVA. Only group iv showed a significant decrease in total cell number, indicating that at 10 µM BPIP may negatively influence embryo development. At both 5 and 10 µM, BPIP treatment resulted in an increase in percentage of apoptotic cells compared to the control group. However, a similar increase was observed using DMSO alone (group ii), indicating that the apoptotic effect may be due solely to the DMSO. In conclusion, BPIP does not appear to cause embryo toxicity at 5 µM, but an alternative, less toxic, dissolving agent may be considered. Table 1.Embryotoxicity assay of BPIP


2007 ◽  
Vol 17 (04) ◽  
pp. 761-795 ◽  
Author(s):  
TIMOTHY DEIS ◽  
JOHN MEAKIN ◽  
G. SÉNIZERGUES

It is known that the problem of determining consistency of a finite system of equations in a free group or a free monoid is decidable, but the corresponding problem for systems of equations in a free inverse monoid of rank at least two is undecidable. Any solution to a system of equations in a free inverse monoid induces a solution to the corresponding system of equations in the associated free group in an obvious way, but solutions to systems of equations in free groups do not necessarily lift to solutions in free inverse monoids. In this paper, we show that the problem of determining whether a solution to a finite system of equations in a free group can be extended to a solution of the corresponding system in the associated free inverse monoid is decidable. We are able to use this to solve the consistency problem for certain classes of single-variable equations in free inverse monoids.


1986 ◽  
Vol 29 (1) ◽  
pp. 69-73 ◽  
Author(s):  
M. H. Albert ◽  
J. Lawrence

The Ehrenfeucht Conjecture [5] states that if Μ is a finitely generated free monoid with nonempty subset S, then there is a finite subset T⊂S (a “test set”) such that given two endomorphisms f and g on Μ, f and g agree on S if and only if they agree on T. In[4], the authors prove that the above conjecture is equivalent to the following conjecture: a system of equations in a finite number of unknowns in Μ is equivalent to a finite subsystem. Since a finitely generated free monoid embeds naturally into the free group with the same number of generators, it is natural to ask whether a free group of finite rank has the above property on systems of equations. A restatement of the question motivates the following.


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