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Some results on polish groups

Paolini Gianluca, Shelah Saharon

Reports on Mathematical Logic

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2020

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Vol. 55

61--71

EN

We prove that no quantifier-free formula in the language of group theory can define the ℵ1-half graph in a Polish group, thus generalising some results from [6]. We then pose some questions on the space of groups of automorphisms of a given Borel complete class, and observe that this space must contain at least one uncountable group. Finally, we prove some results on the structure of the group of automorphisms of a locally finite group: firstly, we prove that it is not the case that every group of automorphisms of a graph of power λ is the group of automorphism of a locally finite group of power λ; secondly, we conjecture that the group of automorphisms of a locally finite group of power λ has a locally finite subgroup of power λ, and reduce the problem to a problem on p-groups, thus settling the conjecture in the case λ = ℵ0.

2

Automorphisms of Toeplitz B-free systems

Dymek A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2017

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Vol. 65, no. 2

139--152

EN

For each B-free subshift given by B={2ibi} i ∈ N, where {bi} i ∈ N is a set of pairwise coprime odd numbers greater than one, it is shown that the automorphism group of the subshift consists solely of powers of the shift.

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On the family of cubical multivariate cryptosystems based on the algebraic graph over finite commutative rings of characteristic 2

Romańczuk U., Ustimenko V.

Annales Universitatis Mariae Curie-Skłodowska. Sectio AI, Informatica

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2012

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Vol. 12, no. 3

89--106

EN

The family of algebraic graphs A(n;K) defined over the finite commutative ring K were used for the design of different multivariate cryptographical algorithms (private and public keys, key exchange protocols). The encryption map corresponds to a special walk on this graph. We expand the class of encryption maps via the use of an automorphism group of A(n;K). In the case of characteristic 2 the encryption transformation is a Boolean map. We change finite field for the commutative ring of characteristic 2 and consider some modifications of algorithm which allow to hide a ground commutative ring.

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Automorphisms of Witt rings of finite fields

Stępień M. R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2011

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Vol. 16

67--70

EN

The problem of general description of the group of automorphisms of any Witt ring W seems to be very difficult to solve. However, there are many types of Witt rings, which automorphism are described precisely (e.g. [1], [2], [4], [5], [6],[7], [8]). In our paper we characterize automorphisms of abstract Witt rings (cf. [3]) isomorphic to powers of Witt rings of quadratic forms with coefficients in finite fields with characteristic different from 2.

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Ultra regular covering space and its automorphism group

Han S. E.

International Journal of Applied Mathematics and Computer Science

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2010

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Vol. 20, no 4

699-710

EN

In order to classify digital spaces in terms of digital-homotopic theoretical tools, a recent paper by Han (2006b) (see also the works of Boxer and Karaca (2008) as well as Han (2007b)) established the notion of regular covering space from the viewpoint of digital covering theory and studied an automorphism group (or Deck's discrete transformation group) of a digital covering. By using these tools, we can calculate digital fundamental groups of some digital spaces and classify digital covering spaces satisfying a radius 2 local isomorphism (Boxer and Karaca, 2008; Han, 2006b; 2008b; 2008d; 2009b). However, for a digital covering which does not satisfy a radius 2 local isomorphism, the study of a digital fundamental group of a digital space and its automorphism group remains open. In order to examine this problem, the present paper establishes the notion of an ultra regular covering space, studies its various properties and calculates an automorphism group of the ultra regular covering space. In particular, the paper develops the notion of compatible adjacency of a digital wedge. By comparing an ultra regular covering space with a regular covering space, we can propose strong merits of the former.

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Theorems on products of central collineations with distinct centers or axes applied to the Benz planes

Jakóbowski J.

Demonstratio Mathematica

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2004

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Vol. 37, nr 3

639--653

EN

There are three kinds of the Benz planes: Mobius planes, Laguerre planes and Minkowski planes [2, 3, 7]. In any Benz plane an automorphism φ is central if φ has a fixed point P and becomes a central collineation in the projective derived plane induced by P. Such central automorphisms have been considered by many authors (cf. [8,13, 11, 12, 10]), in particular the automorphism groups were classified. Usually product of two central collineations without common center or common axis is not central. But in some special cases it is central [4]. In this paper we apply theorems concerning such special cases - to the Benz planes.