On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group, π _e (G) denotes the set of orders of elements in G. If Ω is a subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups with the same set Ω of element orders. We say that G is k-distinguishable if h(π _e (G)) = k < ∞, otherwise G is called non-distinguishable. Usually, a 1-distinguishable group is called a characterizable group. It is shown that if M is a sporadic simple group different from M ₁₂, M ₂₂, J ₂, He, Suz, M ^c L and O′N, then Aut(M) is characterizable by its element orders. It is also proved that if M is isomorphic to M ₁₂, M ₂₂, He, Suz or O′N, then h(π _e (Aut(M))) ∈¸ {1,∞}.     

图的可定向嵌入的标根可数性

给定一族图G,可定向曲面上存在多少个以其中某个图为基础图的标根地图？采用图的自同构群对图在可定向曲面上的嵌入集合进行分类，该文解决了这个问题，同时得到了求解计数函数f^r(M)的一种新的方法。     

C(X) A上的自同构群

本文研究了连续函数代数C(X)与某个C*-代数 A的张量积C(X) A的自同构群.当 A是有单位元且具有平凡中心的C*-代数时,本文完全刻划了C(X) A的自同构群.利用AF-代数的K-理论,本文还刻划了当X是全不连通的紧致Hausdorff空间时,C(X)与紧算子理想的张量积的自同构群.     

The automorphism group of Payne derived generalized quadrangles

We solve a long-standing open problem by proving that the automorphism group of any thick Payne derived generalized quadrangle with ambient quadrangle S a thick generalized quadrangle of order s, s?5 and odd, with a center of symmetry, is induced by the automorphism group of S.     

On outer automorphism groups of coxeter groups

Summary It is shown that the outer automorphism group of a Coxeter groupW of finite rank is finite if the Coxeter graph contains no infinite bonds. A key step in the proof is to show that if the group is irreducible andΠ ₁ andΠ ₂ any two bases of the root system ofW, thenΠ ₂ = ±ωΠ ₁ for some ω εW. The proof of this latter fact employs some properties of the dominance order on the root system introduced by Brink and Howlett. This article was processed by the author using the Springer-Verlag TEX PJour1g macro package 1991.     

Automorphisms of the endomorphism semigroup of a free commutative dimonoid

We determine all isomorphisms between the endomorphism semigroups of free commutative dimonoids and prove that all automorphisms of the endomorphism semigroup of a free commutative dimonoid are quasi-inner. In particular, we answer a question of B. I. Plotkin.     

Structure of relatively free dimonoids

Loday introduced the notion of a dimonoid and constructed the free dimonoid. The paper concerns the variety theory of dimonoids. We recall and summarize the results obtained by Loday, the author as well as others on the structure of some relatively free dimonoids. This part of the paper should be viewed as a survey, however we include some new ideas in the last sections. Namely, we construct a free (?z;rs)-dimonoid, a free (rs;rz)-dimonoid, a free rs-dimonoid, a free (rb;rs)-dimonoid, a free (rs;rb)-dimonoid and characterize the least (?z;rs)-congruence, the least (rs;rz)-congruence, the least rs-congruence, the least (rb;rs)-congruence and the least (rs;rb)-congruence on the free dimonoid. We also establish that the automorphism groups of constructed free algebras are isomorphic to the symmetric group and the semigroups of the free rs-dimonoid are anti-isomorphic.     

On quartic half-arc-transitive metacirculants

Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ρ and σ, where ρ is (m,n)-semiregular for some integers m≥1, n≥2, and where σ normalizes ρ, cyclically permuting the orbits of ρ in such a way that σ ^m has at least one fixed vertex. A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic half-arc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed. Both authors were supported in part by “ARRS – Agencija za znanost Republike Slovenije”, program no. P1-0285.