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Identities with generalized skew derivations on Lie ideals
Vincenzo De Filippis, Ajda Fošner, Feng Wei, 2013, original scientific article

Abstract: Let ▫$m, n$▫ be two nonzero fixed positive integers, ▫$R$▫ a 2-torsion free prime ring with the right Martindale quotient ring ▫$Q$▫, ▫$L$▫ a non-central Lie ideal of ▫$R$▫, and ▫$\delta$▫ a derivation of ▫$R$▫. Suppose that ▫$\alpha$▫ is an automorphism of ▫$R$▫, ▫$D$▫ a skew derivation of ▫$R$▫ with the associated automorphism ▫$\alpha$▫, and ▫$F$▫ a generalized skew derivation of ▫$R$▫ with the associated skew derivation ▫$D$▫. If ▫$$F(x^{m+n}) = F(x^m)x^n + x^m \delta (x^n)$$▫ is a polynomial identity for ▫$L$▫, then either ▫$R$▫ satisfies the standard polynomial identity ▫$s_4(x_1, x_2, x_3, x_4)$▫ of degree 4, or ▫$F$▫ is a generalized derivation of ▫$R$▫ and ▫$\delta = D$▫. Furthermore, in the latter case one of the following statements holds: (1) ▫$D = \delta = 0$▫ and there exists ▫$a \in Q$▫ such that ▫$F(x) = ax$▫ for all ▫$x \in R$▫; (2) ▫$\alpha$▫ is the identical mapping of ▫$R$▫.
Found in: osebi
Keywords: mathematics, algebra, polynomial identity, generalized skew derivation, prime ring
Published: 15.10.2013; Views: 1924; Downloads: 90
URL Full text (0,00 KB)

On 2-fold covers of graphs
Yan-Quan Feng, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2008, original scientific article

Abstract: A regular covering projection ▫$\wp : \widetilde{X} \to X$▫ of connected graphs is ▫$G$▫-admissible if ▫$G$▫ lifts along ▫$\wp$▫. Denote by ▫$\tilde{G}$▫ the lifted group, and let CT▫$(\wp)$▫ be the group of covering transformations. The projection is called ▫$G$▫-split whenever the extension ▫{$\mathrm{CT}}(\wp) \to \tilde{G} \to G$▫ splits. In this paper, split 2-covers are considered, with a particular emphasis given to cubic symmetric graphs. Supposing that ▫$G$▫ is transitive on ▫$X$▫, a ▫$G$▫-split cover is said to be ▫$G$▫-split-transitive if all complements ▫$\tilde{G} \cong G$▫ of CT▫$(\wp)$▫ within ▫$\tilde{G}$▫ are transitive on ▫$\widetilde{X}$▫; it is said to be ▫$G$▫-split-sectional whenever for each complement ▫$\tilde{G}$▫ there exists a ▫$\tilde{G}$▫-invariant section of ▫$\wp$▫; and it is called ▫$G$▫-split-mixed otherwise. It is shown, when ▫$G$▫ is an arc-transitive group, split-sectional and split-mixed 2-covers lead to canonical double covers. Split-transitive covers, however, are considerably more difficult to analyze. For cubic symmetric graphs split 2-cover are necessarily canonical double covers (that is, no ▫$G$▫-split-transitive 2-covers exist) when ▫$G$▫ is 1-regular or 4-regular. In all other cases, that is, if ▫$G$▫ is ▫$s$▫-regular, ▫$s=2,3$▫ or ▫$5$▫, a necessary and sufficient condition for the existence of a transitive complement ▫$\tilde{G}$▫ is given, and moreover, an infinite family of split-transitive 2-covers based on the alternating groups of the form ▫$A_{12k+10}$▫ is constructed. Finally, chains of consecutive 2-covers, along which an arc-transitive group ▫$G$▫ has successive lifts, are also considered. It is proved that in such a chain, at most two projections can be split. Further, it is shown that, in the context of cubic symmetric graphs, if exactly two of them are split, then one is split-transitive and the other one is either split-sectional or split-mixed.
Found in: osebi
Keywords: graph theory, graphs, cubic graphs, symmetric graphs, ▫$s$▫-regular group, regular covering projection
Published: 15.10.2013; Views: 1633; Downloads: 15
URL Full text (0,00 KB)

Existence of non-Cayley Haar graphs
István Kovács, Yan-Quan Feng, Jie Wang, Da-Wei Yang, 2020, original scientific article

Found in: osebi
Keywords: graph, Cayley graph, Haar graph
Published: 17.06.2020; Views: 65; Downloads: 3
URL Full text (0,00 KB)

Elementary abelian groups of rank 5 are DCI-groups
István Kovács, Yan-Quan Feng, 2018, published scientific conference contribution abstract (invited lecture)

Found in: osebi
Keywords: elementary abelian group, rank, DCI-group
Published: 07.02.2018; Views: 845; Downloads: 57
URL Full text (0,00 KB)

On groups all of whose Haar graphs are Cayley graphs
Da-Wei Yang, István Kovács, Yan-Quan Feng, 2019, original scientific article

Found in: osebi
Keywords: graph automorphism, Cayley graph, Haar graph
Published: 28.06.2019; Views: 491; Downloads: 220
URL Full text (0,00 KB)
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On cubic symmetric non-Cayley graphs with solvable automorphism groups
Yan-Quan Feng, Klavdija Kutnar, Dragan Marušič, Da-Wei Yang, 2019, original scientific article

Found in: osebi
Keywords: symmetric graph, non-Cayley graph, regular cover
Published: 18.11.2019; Views: 486; Downloads: 30
URL Full text (0,00 KB)

On isomorphism and multiplier inequivalence of cyclic steiner quadruple systems
Edward Dobson, Tao Feng, Derek F. Holt, Patric R. J. Östergård, 2018, published scientific conference contribution abstract (invited lecture)

Found in: osebi
Keywords: isomorphism, cyclic Steiner quadruple system, automorphism
Published: 07.02.2018; Views: 1331; Downloads: 11
URL Full text (0,00 KB)

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