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Unitary and Symmetric Units of a Commutative Group Algebra

Part of: Rings and algebras arising under various constructions Representation theory of groups Conditions on elements

Published online by Cambridge University Press:  27 December 2018

V. A. Bovdi  and
A. N. Grishkov
V. A. Bovdi*
Affiliation:
UAEU, Al-Ain, United Arab Emirates (vbovdi@gmail.com)
A. N. Grishkov
Affiliation:
IME USP, Citade Universitària, Sao Paulo, Brazil (shuragri@gmail.com) and Omsk F.M. Dostoevsky State University, Omsk Russia
*
*Corresponding author.
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Abstract

Let F be a field of characteristic two and G a finite abelian 2-group with an involutory automorphism η. If G = H × D with non-trivial subgroups H and D of G such that η inverts the elements of H (H without a direct factor of order 2) and fixes D element-wise, then the linear extension of η to the group algebra FG is called a nice involution. This determines the groups of unitary and symmetric normalized units of FG. We calculate the orders and the invariants of these subgroups.

Keywords

group ringunitary groupsymmetric elementcommutative ringinvolution

MSC classification

Primary: 16S34: Group rings , Laurent polynomial rings 16U60: Units, groups of units
Secondary: 20C05: Group rings of finite groups and their modules
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 3 , August 2019 , pp. 641 - 654
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Abdukhalikov, V., Defining sets of extended cyclic codes invariant under the affine group, J. Pure Appl. Algebra 196(1) (2005), 1–19.Google Scholar
2.Abdukhalikov, V., On codes over rings invariant under affine groups, Adv. Math. Commun. 7(3) (2013), 253–265.Google Scholar
3.Aĭzenberg, N. N., Bovdi, A. A., Gergo, È. I. and Geche, F. È., Algebraic aspects of threshold logic (in Russian, with English summary), Cybernetics 16(2) (1980), 188–193.Google Scholar
4.Balogh, Z., Creedon, L. and Gildea, J., Involutions and unitary subgroups in group algebras, Acta Sci. Math. (Szeged) 79(3–4) (2013), 391–400.Google Scholar
5.Barakat, M., Computations of unitary groups in characteristic 2, (for J.-P. Serre), preprint, 2013.Google Scholar
6.Berman, S. D. and Grushko, I. I., B-functions encountered in modular codes, Problemy Peredachi Informatsii 17(2) (1981), 10–18 (in Russian).Google Scholar
7.Bovdi, V. and Kovács, L. G., Unitary units in modular group algebras, Manuscripta Math. 84(1) (1994), 57–72.Google Scholar
8.Bovdi, A. A. and Sakach, A. A., The unitary subgroup of the multiplicative group of the modular group algebra of a finite abelian p-group, Mat. Zametki 45(6) (1989), 23–29, 110.Google Scholar
9.Bovdi, V. and Salim, M., On the unit group of a commutative ring, Acta Sci. Math. (Szeged) 80(3–4) (2014), 434–445.Google Scholar
10.Bovdi, A. A. and Szakács, A., A basis for the unitary subgroup of the group of units in a finite commutative group algebra, Publ. Math. Debrecen 46(1–2) (1995), 97–120.Google Scholar
11.Bovdi, A. and Szakács, A., Units of commutative group algebra with involution, Publ. Math. Debrecen 69(3) (2006), 291–296.Google Scholar
12.Gildea, J., The structure of the unitary units of the group algebra $\Bbb F_{2^{k}}D_{8}$, Int. Electron. J. Algebra 9 (2011), 171–176.Google Scholar
13.Hurley, B. and Hurley, T., Group ring cryptography, Int. J. Pure Appl. Math. 69(1) (2011), 67–86.Google Scholar
14.Hurley, B. and Hurley, T., Paraunitary matrices and group rings, Int. J. Group Theor. 3(1) (2014), 31–56.Google Scholar
15.Novikov, S. P., Algebraic construction and properties of Hermitian analogs of K-theory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and the theory of characteristic classes. I. II, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 253–288, 475–500.Google Scholar
16.Sandling, R., Units in the modular group algebra of a finite abelian p-group, J. Pure Appl. Algebra 33(3) (1984), 337–346.Google Scholar
17.Serre, J.-P., Bases normales autoduales et groupes unitaires en caractéristique 2, Transform. Groups 19(2) (2014), 643–698.Google Scholar
18.Willems, W., A note on self-dual group codes, IEEE Trans. Inform. Theory 48(12) (2002), 3107–3109.Google Scholar