On two problems from the kourovka notebook

Translated from Matematicheskie Zametki, Vol. 55, No. 6, pp. 59–63, June, 1994.     

A problem from the kourovka notebook on embeddings of the group \mathbb{Q}

Answering the question of de la Harpe and Bridson in the Kourovka Notebook, Problem 14.10(b), we construct explicit embeddings of the additive group of rational numbers $ \mathbb{Q} $ in a finitely generated group G. In fact, the group G is 2-generator, and the constructed embedding can be subnormal and preserve a few properties such as solubility or torsion freeness.     

On one question of B. Amberg

In the case where a group G is a productG=AB of Abelian subgroupsA andB one of which has a finite 0-rank, we prove that the Fitting subgroupF and the Hirsch-Plotkin radicalR admit the decompositionsF=(F A)(F B) andR=(RA)(R B), respectively. This gives the affirmative answer to one question of B. Amberg.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 457–461, April, 1994.This paper was supported by the Ukrainian State Foundation on Science and Technology.     

On the variety question for wRRA

In this paper we consider a question of Jónsson [6] whether the class of weakly representable relation algebras is a variety. We prove that the class is closed under taking homomorphic images provided that a certain embedding condition obtains. Received June 21, 2005; accepted in final form October 17, 2006.     

On a question ofMakar-Limanov

Let be an uncountable field, let be a field extension, and let be an associative -algebra. We show that if contains a non-commutative free algebra, then so does .

     

On a question of Pereira

Answering a question of Pereira we show that it is possible to have a model violating the Singular Cardinal Hypothesis without a tree-like continuous scale. We are grateful to Luis Pereira for asking an interesting question and his comments on the paper. We would like to thank to the referee of the paper for his comments and corrections.     

On a question of Goss

In this note we answer the question raised by D. Goss in [D. Goss, Applications of non-Archimedean integration to the L-series of τ-sheaves, J. Number Theory 110 (1) (2005) 83-113] by proving that the group of locally analytic endomorphisms on the 1-units of a locally compact field of characteristic p>0 is isomorphic to the p-adic integers.     

On a question of Gross

Using the notion of weighted sharing of sets we prove two uniqueness theorems which improve the results proved by Fang and Qiu [H. Qiu, M. Fang, A unicity theorem for meromorphic functions, Bull. Malaysian Math. Sci. Soc. 25 (2002) 31-38], Lahiri and Banerjee [I. Lahiri, A. Banerjee, Uniqueness of meromorphic functions with deficient poles, Kyungpook Math. J. 44 (2004) 575-584] and Yi and Lin [H.X. Yi, W.C. Lin, Uniqueness theorems concerning a question of Gross, Proc. Japan Acad. Ser. A 80 (2004) 136-140] and thus provide an answer to the question of Gross [F. Gross, Factorization of meromorphic functions and some open problems, in: Proc. Conf. Univ. Kentucky, Lexington, KY, 1976, in: Lecture Notes in Math., vol. 599, Springer, Berlin, 1977, pp. 51-69], under a weaker hypothesis.     

On a question of Zhang

When studying the conjugacy class version of the Huppert’s ρ-σ conjecture, Jiping Zhang raised a number theory question. In this article, we provide examples to show that the question raised by Zhang is not always true in general.     

On the Chatyrko-Hattori solution of de Groot's question

Let Zn=(n[0,1]×(0,1])∪(∂(n[0,1])×{0}). De Groot asked: Is cmpZn?n for every n? It is known that the answer is yes for n=1 and 2. V.A. Chatyrko and Y. Hattori [Fund. Math. 172 (2002) 107-115] showed that the answer is no for n?5. It is shown that the answer is also no for n=4. The question is unresolved for n=3.