Searcharxiv⌕ Search

arXiv subjects

Aleksandr Golubchik

Publications and source records attributed to Aleksandr Golubchik.

6 recordsLinked to original sources
arXiv2007

The graph isomorphism problem is polynomial

It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties of regular $k$-partitions that, on one hand, generalize automorphic $k$-partitions (=systems of $k$-orbits of permutation groups), and, on other hand, schemes of relations (strongly regular 2-partitions or regular 3-partitions), that are a subject of the algebraic combinatorics. It is shown that the stabilization of a graph by quadrangles detects the triviality of the graph automorphism group. The result is obtained by lineariation of the algebraic combinatorics. Keywords: $k$-partitions, symmetry, algebraic combinatorics

math.GM↗
arXiv2005

The $k$-orbit theory and Fein-Kantor-Schacher Theorem

By the investigation of $k$-orbits symmetry properties it is obtained a simple proof of the B. Fein, W. M. Kantor and M. Schacher Theorem: any transitive permutation group contains a non-trivial fixed-point-free prime-power element. Key words: $k$-orbits, partitions, permutations, symmetry, groups

math.GM↗
arXiv2005

The $k$-orbit theory and polycirculant conjecture

The paper contains a proof of the conjecture of M. Klin and D. Maru$\breve{\rm s}$i$\breve{\rm c}$ that an automorphism group of a transitive graph contains a permutation, decomposed in cycles of the same length. The proof is based on the $k$-orbit theory developed by author. Key words: $k$-orbits, partitions, permutations, symmetry, groups

math.GM↗
arXiv2004

On the nature of finite groups

The reality of the difficulties in investigation of finite groups are considered. It is shown that the consideration of symmetry properties of the $k$-orbits that are obtained with an action of a finite group $F=(V,\cdot)$ on Cartesian power $V^k$ gives a new view on the nature of groups and simplifies some difficult properties of groups. Using this representation it is obtained a simple proof of the W. Feit, J.G. Thompson theorem: Solvability of groups of odd order.

math.GM↗
arXiv2003

On the polycirculant conjecture

In the paper the foundation of the $k$-orbit theory is developed. The theory opens a new simple way to the investigation of groups and multidimensional symmetries. The relations between combinatorial symmetry properties of a $k$-orbit and its automorphism group are found. It is found the local property of a $k$-orbit. The difference between 2-closed group and $m$-closed group for $m>2$ is discovered. It is explained the specific property of Petersen graph automorphism group $n$-orbit. It is shown that any non-trivial primitive group contains a transitive imprimitive subgroup and as a result it is proved that the automorphism group of a vertex transitive graph (2-closed group) contains a regular element (polycirculant conjecture). Using methods of the $k$-orbit theory, it is considered different possibilities of permutation representation of a finite group and shown that the most informative, relative to describing of the structure of a finite group, is the permutation representation of the lowest degree. Using this representation it is obtained a simple proof of the W. Feit, J.G. Thompson theorem: Solvability of groups of odd order. It is described the enough simple structure of lowest degree representation of finite groups and found a way to constructing of the simple full invariant of a finite group. To the end, using methods of $k$-orbit theory, it is obtained one of possible polynomial solutions of the graph isomorphism problem.

math.GM↗