We prove that the statement ‘For all Borel ideals I and J on ω, every isomorphism between Boolean algebras P(ω)/I and P(ω)/J has a continuous representation’ is relatively consistent with ZFC. In this model every isomorphism between P(ω)/I and any other quotient P(ω)/J over a Borel ideal is trivial for a number of Borel ideals I on ω. We can also assure that the dominating number, d, is equal t...

An automorphism σ of a simple finite dimensional complex Lie algebra g is called torsion, if σ has finite order in the group Aut(g) of all automorphisms of g. The torsion automorphisms of g were classified by Victor Kac in [12], as an application of his results on infinite dimensional Lie algebras. Those torsion automorphisms contained in the identity component G = Aut(g)◦ are called inner; the...

Let $G$ be a $p$-group of order $p^n$ and $Phi$=$Phi(G)$ be the Frattini subgroup of $G$. It is shown that the nilpotency class of $Autf(G)$, the group of all automorphisms of $G$ centralizing $G/ Fr(G)$, takes the maximum value $n-2$ if and only if $G$ is of maximal class. We also determine the nilpotency class of $Autf(G)$ when $G$ is a finite abelian $p$-group.

Abstract We describe the endomorphisms of direct product two free groups finite rank and obtain conditions for which subgroup fixed points is finitely generated, we do same periodic points. also automorphisms such a group solve three Whitehead problems automorphisms, monomorphisms in this class. an endomorphism to be uniformly continuous when suitable metric considered study infinite extension ...

We study those fully irreducible outer automorphisms φ of a finite rank free group Fr which are parageometric, meaning that the attracting fixed point of φ in the boundary of outer space is a geometric R-tree with respect to the action of Fr, but φ itself is not a geometric outer automorphism in that it is not represented by a homeomorphism of a surface. Our main result shows that the expansion...

The characterization of automorphisms of Bernstein algebras is an open problem. We only know some particular results. Previously we have characterized the automorphisms of quasiorthogonal, orthogonal, and strongly orthogonal weak Bernstein-Jordan algebras. In this paper we work on the minimal dimension with respect to quasiorthogonality, orthogonality, and strong orthogonality. We establish som...

We prove a compactness theorem for automorphisms of free groups. Namely, we show that the set of automorphisms keeping bounded the length of the uniform current is compact (up to conjugation.) This implies that the spectrum of the length of the images of the uniform current is discrete, answering to a conjecture of I. Kapovich.

In this work we introduce the concept of Generalized Atanassov’s Intuitionistic Fuzzy Index. We characterize it in terms of fuzzy implication operators and we propose a construction method with automorphisms. Finally, we study some special properties of the generalized Atanassov’s intuitionistic fuzzy index. Keywords— Atanassov’s Intuitionistic Fuzzy Index, Fuzzy Implication Operator, Automorph...

We show that the natural representation of SL 3 SL 5 SL 13 allows nonlinear equivariant automorphisms; more exactly, the group of polynomial au-tomorphisms on C 3 C 5 C 13 commuting with the simple SL 3 SL 5 SL 13-action is isomorphic to C n C. This is the rst example of a simple module with nonlinear equivariant automorphisms.

Among other things, we prove that the group of automorphisms fixing every normal subgroup of a (nilpotent of class c)-by-abelian group is (nilpotent of class ≤ c)-by-metabelian. In particular, the group of automorphisms fixing every normal subgroup of a metabelian group is soluble of derived length at most 3. An example shows that this bound cannot be improved.