Thin-tall Boolean algebras and their automorphism groups

Our main result is a general construction to show that for each countable groupG there are 2¹ pairwise non-isomorphic thin-tall superatomic Boolean algebras, all of them representingG as a group of non-trivial automorphisms.Presented by J. Mycielski.Research supported by the NSERC of Canada.     

On the automorphism groups of countable Boolean algebras

We prove categoricity results for the class {<B, Aut(B)>|B is a countable BA}. This paper is part of the author’s doctoral dissertation prepared at the Hebrew University under the supervision of Professor Saharon Shelah.     

Structure of weak automorphism groups of mono-unary algebras

A weak automorphism of an algebra on A with the set T of term operations is a permutation of of A such that . In this note we describe the groups of weak automorphisms of mono-unary algebras. Some examples of such algebras without weak automorphisms are given. Received January 31, 1996; accepted in final form March 16, 1999.     

The discriminant controls automorphism groups of noncommutative algebras

We use the discriminant to determine the automorphism groups of some noncommutative algebras, and we prove that a family of noncommutative algebras has tractable automorphism groups.     

Cluster automorphism groups of cluster algebras with coefficients

We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. We introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra (i.e., the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients, cluster algebras with universal geometric coefficients, and cluster algebras from surfaces (except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.     

Saturated boolean algebras and their stone spaces

For an infinite cardinal κ, we call a compact zero-dimensional space a κ-Parovicenko space if its boolean algebra of clopen sets is κ-saturated and has cardinality κ^<κ. We answer some questions about these spaces which were posed in [14]. For instance, it is shown that a κ⁺-Parovicenko spacé can be a Stone-Cech remainder in a natural way. We show that some of the results in [14] which used the assumption κ^<κ=κ, do indeed require this assumption. We also show that if 2^κ=κ⁺ then each compact F_κ⁺-space with weight κ⁺ can be embedded into a κ⁺-Parovicenko space (and so into an extremally disconnected space).     

On automorphism groups of connected Lie groups

We prove that ifG is a connected Lie group with no compact central subgroup of positive dimension then the automorphism group ofG is an almost algebraic subgroup of , where is the Lie algebra ofG. We also give another proof of a theorem of D. Wigner, on the connected component of the identity in the automorphism group of a general connected Lie group being almost algebraic, and strengthen a result of M.Goto on the subgroup consisting of all automorphisms fixing a given central element.     

On automorphism groups of some finite groups

We show that if n > 1 is odd and has no divisor p4 for any prime p, then there is no finite group G such that│Aut(G)│ = n.     

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.