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The structure of the unit group of the group algebra of Pauli’s group over any field of characteristic 2. (English) Zbl 1205.16031

Let \(P\) be the Pauli group of order 16, that is, the central product of the quaternion group and a cyclic group of order 4, \(F_{2^k}P\) the group algebra over the Galois-field and \(U\) its group of units. The author determines the structure of the group \(U\) as follows: \(((((C_2^{9k}\times C_4^k)\rtimes C_4^k)\rtimes C_2^k)\rtimes C_2^k)\times C_{2^k-1}\). These calculations are done by means of the matrix representation described by T. Hurley, [Int. J. Pure Appl. Algebra 31, No. 3, 319-335 (2006; Zbl 1136.20004)], and continue the research on the structure of the group of units by L. Creedon, J. Gildea, [Int. J. Algebra Comput. 19, No. 2, 283-286 (2009; Zbl 1171.16302)].

MSC:

16U60 Units, groups of units (associative rings and algebras)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
15B33 Matrices over special rings (quaternions, finite fields, etc.)
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References:

[1] Bovdi A. A., Math. Zametki 45 pp 23–
[2] DOI: 10.1007/BF02567443 · Zbl 0816.16027 · doi:10.1007/BF02567443
[3] DOI: 10.1080/00927870008826934 · Zbl 0952.16022 · doi:10.1080/00927870008826934
[4] Creedon L., Can. Math. Bull.
[5] DOI: 10.1142/S0218196709005081 · Zbl 1171.16302 · doi:10.1142/S0218196709005081
[6] Davis P. J., Circulant Matrices (1979)
[7] Hurley T., Int. J. Pure Appl. Math. 31 pp 319–
[8] DOI: 10.1007/978-94-010-0405-3 · doi:10.1007/978-94-010-0405-3
[9] DOI: 10.1016/0022-4049(84)90066-5 · Zbl 0543.20008 · doi:10.1016/0022-4049(84)90066-5
[10] Sandling R., Math. Comp. 59 pp 689–
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