Units of the group algebra of the group C_n x D_6 over any finite field of characteristic 3
Abstract
In this paper, we establish the structure of the unit group of the group algebra ${\FF}_{3^t}(C_n\times D_6)$ for $n \geq 1$.Citation
Gildea, J., & Taylor, R. (2018). Units of the group algebra of the group C_n x D_6 over any finite field of characteristic 3. International Electronic Journal of Algebra, 24.Additional Links
http://www.ieja.net/Type
ArticleLanguage
enEISSN
1306- 6048ae974a485f413a2113503eed53cd6c53
10.24330/ieja.440205
Scopus Count
Collections
Related items
Showing items related by title, author, creator and subject.
-
Torsion Units for a Ree group, Tits group and a Steinberg triality groupGildea, Joe; University of Chester (Springer, 2015-12-28)We investigate the Zassenhaus conjecture for the Steinberg triality group ${}^3D_4(2^3)$, Tits group ${}^2F_4(2)'$ and the Ree group ${}^2F_4(2)$. Consequently, we prove that the Prime Graph question is true for all three groups.
-
Group Codes, Composite Group Codes and Constructions of Self-Dual CodesGildea, Joe; Korban, Adrian (University of Chester, 2021-08)The main research presented in this thesis is around constructing binary self-dual codes using group rings together with some well-known code construction methods and the study of group codes and composite group codes over different alphabets. Both these families of codes are generated by the elements that come from group rings. A search for binary self-dual codes with new weight enumerators is an ongoing research area in algebraic coding theory. For this reason, we present a generator matrix in which we employ the idea of a bisymmetric matrix with its entries being the block matrices that come from group rings and give the necessary conditions for this generator matrix to produce a self-dual code over a fi nite commutative Frobenius ring. Together with our generator matrix and some well-known code construction methods, we find many binary self-dual codes with parameters [68, 34, 12] that have weight enumerators that were not known in the literature before. There is an extensive literature on the study of different families of codes over different alphabets and speci fically finite fi elds and finite commutative rings. The study of codes over rings opens up a new direction for constructing new binary self-dual codes with a rich automorphism group via the algebraic structure of the rings through the Gray maps associated with them. In this thesis, we introduce a new family of rings, study its algebraic structure and show that each member of this family is a commutative Frobenius ring. Moreover, we study group codes over this new family of rings and show that one can obtain codes with a rich automorphism group via the associated Gray map. We extend a well established isomorphism between group rings and the subring of the n x n matrices and show its applications to algebraic coding theory. Our extension enables one to construct many complex n x n matrices over the ring R that are fully de ned by the elements appearing in the first row. This property allows one to build generator matrices with these complex matrices so that the search field is practical in terms of the computational times. We show how these complex matrices are constructed using group rings, study their properties and present many interesting examples of complex matrices over the ring R. Using our extended isomorphism, we de ne a new family of codes which we call the composite group codes or for simplicity, composite G-codes. We show that these new codes are ideals in the group ring RG and prove that the dual of a composite G-code is also a composite G-code. Moreover, we study generator matrices of the form [In | Ω(v)]; where In is the n x n identity matrix and Ω(v) is the composite matrix that comes from the extended isomorphism mentioned earlier. In particular, we show when such generator matrices produce self-dual codes over finite commutative Frobenius rings. Additionally, together with some generator matrices of the type [In | Ω(v)] and the well-known extension and neighbour methods, we fi nd many new binary self-dual codes with parameters [68, 34, 12]. Lastly in this work, we study composite G-codes over formal power series rings and finite chain rings. We extend many known results on projections and lifts of codes over these alphabets. We also extend some known results on γadic codes over the infi nite ring R∞
-
Re-naming and re-framing: Evolving the ‘Higher Education Research Group’ to the ‘Geography and Education Research Group’Healey, Ruth L.; West, Harry; University of Chester; University of the West of England (Wiley, 2021-10-22)Editorial for special issue of Area about the evolution of the Higher Education Research Group to the Geography and Education Research Group of the Royal Geographical Society.