On the isomorphism of unitary subgroups of noncommutative group algebras

Year 2022, Volume: 9 Issue: 2, 45 - 51, 13.05.2022

Zsolt Adam Balogh

https://doi.org/10.13069/jacodesmath.1111746

Abstract

Let $FG$ be the group algebra of a finite $p$-group $G$ over a field $F$ of characteristic $p$. Let $\text{*}$ be an involution of the group algebra $FG$ which arises form the group basis $G$. The upper bound for the number of non-isomorphic $\text{*}$-unitary subgroups is the number of conjugacy classes of the automorphism group $G$ with all the elements of order two. The upper bound is not always reached in the case when $G$ is an abelian group, but for non-abelian case the question is open. In this paper we present a non-abelian $p$-group $G$ whose group algebra $FG$ has sharply less number of non-isomorphic $\text{*}$-unitary subgroups than the given upper bound.

Keywords

Group ring, Group of units, Unitary subgroup

References

• [1] Z. Balogh, On unitary subgroups of group algebras, Int. Electron. J. Algebra 29(29) (2021) 187–198.
• [2] Z. Balogh, V. Bovdi, The isomorphism problem of unitary subgroups of modular group algebras, Publ. Math. Debrecen 97(1-2) (2020) 27–39.
• [3] Z. Balogh, L. Creedon, J. Gildea, Involutions and unitary subgroups in group algebras, Acta Sci. Math. (Szeged) 79(3-4) (2013) 391–400.
• [4] Z. Balogh, V. Laver. Unitary subgroups of commutative group algebras of characteristic 2. Ukraïn. Mat. Zh. 72(6) (2020) 751–757.
• [5] A. Bovdi, L. Erdei, Unitary units in modular group algebras of groups of order 16, Tech. Rep., Univ. Debrecen, L. Kossuth Univ. 4(157) (1996) 1–16.
• [6] A. Bovdi, L. Erdei. Unitary units in modular group algebras of 2-groups. Comm. Algebra 28(2) (2000) 625–630.
• [7] A. Bovdi, P. Lakatos, On the exponent of the group of normalized units of modular group algebras, Publ. Math. Debrecen 42(3-4) (1993) 409–415.
• [8] A. Bovdi, A. Szakács, Units of commutative group algebra with involution, Publ. Math. Debrecen 69(3) (2006) 291–296.
• [9] A. A. Bovdi, Unitarity of the multiplicative group of an integral group ring, Mat. Sb. (N.S.) 47(2) (1984) 377–389.
• [10] A. A. Bovdi, A. A. Sakach, Unitary subgroup of the multiplicative group of a modular group algebra of a finite abelian p-group. Mat. Zametki 45(6) (1989) 445–450.
• [11] A. A. Bovdi, A.Szakács, 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.
• [12] V. Bovdi, L. G. Kovács, Unitary units in modular group algebras, Manuscripta Math. 84(1) (1994) 57–72.
• [13] V. Bovdi, A. L. Rosa, On the order of the unitary subgroup of a modular group algebra, Comm. Algebra 28(4) (2000) 1897–1905.
• [14] V. Bovdi, M. Salim, On the unit group of a commutative group ring, Acta Sci. Math. (Szeged) 80(3-4) (2014) 433–445.
• [15] V. A. Bovdi, A. N. Grishkov, Unitary and symmetric units of a commutative group algebra, Proc. Edinb. Math. Soc. 62(3) (2019) 641–654.
• [16] L. Creedon, J. Gildea, Unitary units of the group algebra $F_{2^k}D_8$, Internat. J. Algebra Comput. 19(2) (2009) 283–286.
• [17] L. Creedon, J. Gildea, The structure of the unit group of the group algebra$F_{2^k}D_8$, Canad. Math. Bull. 54(2) (2011) 237–243.
• [18] J. Gildea, The structure of the unitary units of the group algebra $F_{2^k}D_8$, Int. Electron. J. Algebra 9 (2011) 171–176.
• [19] D. W. Lewis. Involutions and anti-automorphisms of algebras, Bull. London Math. Soc. 38(4) (2006) 529–545.
• [20] S. P. Novikov, 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, ibid. 34 (1970) 475–500.
• [21] J. -P. Serre, Bases normales autoduales et groupes unitaires en caractéristique 2, (French) [Self-dual normal bases and unitary groups of characteristic 2], Transform. Groups 19(2) (2014) 643–698.
• [22] Y. Wang, H. Liu, The unitary subgroups of group algebras of a class of finite p-groups, J. Algebra Appl. (2021).