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Search: MSC category 20D60 ( Arithmetic and combinatorial problems )

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1. CMB 2010 (vol 54 pp. 39)

Chapman, S. T.; García-Sánchez, P. A.; Llena, D.; Marshall, J.
Elements in a Numerical Semigroup with Factorizations of the Same Length
Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.

Keywords:numerical monoid, numerical semigroup, non-unique factorization
Categories:20M14, 20D60, 11B75

2. CMB 2007 (vol 50 pp. 632)

Zelenyuk, Yevhen; Zelenyuk, Yuliya
Transformations and Colorings of Groups
Let $G$ be a compact topological group and let $f\colon G\to G$ be a continuous transformation of $G$. Define $f^*\colon G\to G$ by $f^*(x)=f(x^{-1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume that $H=\Imag f^*$ is a subgroup of $G$ and for every measurable $C\subseteq H$, $\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable $C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that $f(Sg^{-1})\subseteq Cg^{-1}$ and $\mu(S)\ge(\mu(C))^2$.

Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion,
Categories:05D10, 20D60, 22A10

3. CMB 2004 (vol 47 pp. 530)

Iranmanesh, A.; Khosravi, B.
A Characterization of $ PSU_{11}(q)$
Order components of a finite simple group were introduced in [4]. It was proved that some non-abelian simple groups are uniquely determined by their order components. As the main result of this paper, we show that groups $PSU_{11}(q)$ are also uniquely determined by their order components. As corollaries of this result, the validity of a conjecture of J. G. Thompson and a conjecture of W. Shi and J. Bi both on $PSU_{11}(q)$ are obtained.

Keywords:Prime graph, order component, finite group,simple group
Categories:20D08, 20D05, 20D60

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