Restoration of Information on the Group by the Bottom Layer

Parashchuk, I.A.; Senashov, V. I.

doi:10.31772/2587-6066-2018-19-2-223-226

Cited by 2 publications

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“…A. Parashchuk. As additional restrictions to the task by the bottom layer of the group, the completeness of the group and its layer-finiteness were used [23].…”

Section: The Study Of Simple Non-abelian Groups Using the Bottom Layermentioning

confidence: 99%

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Applications of group theory in crystallography

Senashov

2022

IOP Conf. Ser.: Mater. Sci. Eng.

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Group theory is a powerful tool for studying symmetric physical systems. Such systems include, in particular, molecules and crystals with symmetry. Group theory serves to explain the most important characteristics of atomic spectra. Group theory is also applied to the problems of atomic and nuclear physics. This paper gives examples of the use of the apparatus of group theory in research on crystallography, quantum mechanics, elementary particle physics. In particular, in these studies matrix groups and representations of unitary groups are actively used. For such groups we give an overview of the results on their recognition by the spectrum (by the orders of the elements of the group). This direction has been intensively developed in recent years both in our country and abroad. Recognition of finite simple non-Abelian groups by spectrum has been studied for last thirty years in Yekaterinburg at the Institute of Mathematics and Mechanics of the Ural Division of the Russian Academy of Sciences, in Chelyabinsk Federal University and in the Novosibirsk Institute of Mathematics of Siberian Division of the Russian Academy of Sciences. Some simple non-Abelian groups are not recognizable by their spectra. We have proposed an approach for recognizing groups by the bottom layer. The bottom layer of a group is the set of its elements of prime orders. A group is called recognizable by the bottom layer under additional conditions if it is uniquely restored by the bottom layer under these conditions. The paper considers some examples of simple non-Abelian finite groups that are not recognizable by spectra. For these examples, simultaneous recognition by spectrum and by the bottom layer is proved.

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“…A. Parashchuk. As additional restrictions to the task by the bottom layer of the group, the completeness of the group and its layer-finiteness were used [23].…”

Section: The Study Of Simple Non-abelian Groups Using the Bottom Layermentioning

confidence: 99%

“…G is said to be unrecognizable by the bottom layer if there exist an infinite number of pairwise non-isomorphic groups with the same bottom layer such as in the group G [23].…”

Section: The Study Of Simple Non-abelian Groups Using the Bottom Layermentioning

confidence: 99%

Applications of group theory in crystallography

Senashov

2022

IOP Conf. Ser.: Mater. Sci. Eng.

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On the bottom layer in groups

Senashov

2023

Siberian Aerospace Journal

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The question of the possibility of restoring information about a group by its lower layer, that is, by the set of its elements of prime orders, is considered. The question is classical for mathematical modeling: restoration of the missing information about the object from the part of the preserved data. A group is said to be recognizable from the bottom layer under additional conditions if it is uniquely reconstructed from the bottom layer under these conditions. A group G is said to be almost recognizable from the bottom layer under additional conditions if there exists a finite number of pairwise nonisomorphic groups satisfying these conditions, with the same bottom layer as the group G. A group G is called unrecognizable from the bottom layer under additional conditions if there is an infinite the number of pairwise non-isomorphic groups that satisfy these conditions and have the same bottom layer as the group G. Results are given on the recognition of groups by the bottom layer in various classes of groups. The concept of recognizability by the lower layer was introduced by analogy with the actively studied recognizability by spectrum, that is, by the set of orders of group elements. In this paper, we consider groups that, without a single element, coincide with their bottom layer. Examples of groups with these conditions in the classes of Abelian and non-Abelian groups are given. The properties of such groups are established. The results can be applied when encoding information in space communications.

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Restoration of Information on the Group by the Bottom Layer

Cited by 2 publications

References 0 publications

Applications of group theory in crystallography

Applications of group theory in crystallography

On the bottom layer in groups

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