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Identities with generalized skew derivations on Lie ideals
Ajda Fošner, Vincenzo De Filippis, 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: 3139; Downloads: 138
URL Full text (0,00 KB)

On 2-fold covers of graphs
Klavdija Kutnar, Yan-Quan Feng, 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: 2673; Downloads: 31
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: 1678; Downloads: 73
URL Full text (0,00 KB)

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

Found in: osebi
Keywords: graph automorphism, Cayley graph, Haar graph
Published: 28.06.2019; Views: 1299; Downloads: 327
URL Full text (0,00 KB)
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On cubic symmetric non-Cayley graphs with solvable automorphism groups
Da-Wei Yang, Dragan Marušič, Klavdija Kutnar, Yan-Quan Feng, 2019, original scientific article

Found in: osebi
Keywords: symmetric graph, non-Cayley graph, regular cover
Published: 18.11.2019; Views: 1378; Downloads: 90
URL Full text (0,00 KB)

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

Found in: osebi
Keywords: graph, Cayley graph, Haar graph
Published: 17.06.2020; Views: 701; Downloads: 93
URL Full text (0,00 KB)

On finite groups with prescribed two-generator subgroups and integral Cayley graphs
Yan-Quan Feng, István Kovács, 2021, original scientific article

Found in: osebi
Keywords: 2-group, Frobenius group, special group
Published: 06.05.2021; Views: 322; Downloads: 25
URL Full text (0,00 KB)

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