ON UNITARY SUBGROUPS OF GROUP ALGEBRAS

Year 2021, , 187 - 198, 05.01.2021

Zsolt Adam Balogh

https://doi.org/10.24330/ieja.852199

Cited By: 6

Abstract

Let $FG$ be the group algebra of a finite $p$-group $G$ over a
finite field $F$ of characteristic $p$ and let $*$ be the
classical involution of $FG$. The $*$-unitary subgroup of $FG$,
denoted by $V_*(FG)$, is defined to be the set of all normalized
units $u$ satisfying the property $u^*=u^{-1}$. In this paper we
give a recursive method how to compute the order of the
$*$-unitary subgroup for certain non-commutative group algebras.
A variant of the modular isomorphism question of group algebras is
also considered.

Keywords

Group ring, group of units, unitary subgroup

References

• Z. Balogh and A. Bovdi, Group algebras with unit group of class p, Publ. Math. Debrecen, 65(3-4) (2004), 261-268.
• Z. Balogh and A. Bovdi, On units of group algebras of 2-groups of maximal class, Comm. Algebra, 32(8) (2004), 3227-3245.
• Z. Balogh and V. Bovdi, The isomorphism problem of unitary subgroups of modular group algebras, Publ. Math. Debrecen, 97(1-2) (2020), 27-39, see also arXiv:1908.03877v2 [math.RA].
• Z. Balogh, L. Creedon and J. Gildea, Involutions and unitary subgroups in group algebras, Acta Sci. Math. (Szeged), 79(3-4) (2013), 391-400.
• Z. Balogh and V. Laver, Isomorphism problem of unitary subgroups of group algebras, Ukrainian Math. J., 72(6) (2020), 871-879.
• Z. Balogh and V. Laver, RAMEGA - RAndom MEthods in Group Algebras, Version 1.0.0, (2020).
• S. D. Berman, Group algebras of countable abelian p-groups, Publ. Math. Debrecen, 14 (1967), 365-405.
• A. Bovdi, The group of units of a group algebra of characteristic p, Publ. Math. Debrecen, 52(1-2) (1998), 193-244.
• A. Bovdi and L. Erdei, Unitary units in modular group algebras of groups of order 16, Technical Reports, Universitas Debrecen, Dept. of Math., L. Kossuth Univ., 4(157) (1996), 1-16.
• A. Bovdi and L. Erdei, Unitary units in modular group algebras of 2-groups, Comm. Algebra, 28(2) (2000), 625-630.
• V. A. Bovdi and A. N. Grishkov, Unitary and symmetric units of a commutative group algebra, Proc. Edinb. Math. Soc. (2), 62(3) (2019), 641-654.
• V. Bovdi and L. G. Kovacs, Unitary units in modular group algebras, Manuscripta Math., 84(1) (1994), 57-72.
• V. Bovdi and A. L. Rosa, On the order of the unitary subgroup of a modular group algebra, Comm. Algebra, 28(4) (2000), 1897-1905.
• A. A. Bovdi and 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), 23-29.
• V. Bovdi and M. Salim, On the unit group of a commutative group ring, Acta Sci. Math. (Szeged), 80(3-4) (2014), 433-445.
• A. A. Bovdi and A. Szakacs, 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.
• A. Bovdi and A. Szakacs, Units of commutative group algebra with involution, Publ. Math. Debrecen, 69(3) (2006), 291-296.
• L. Creedon and J. Gildea, Unitary units of the group algebra $\Bbb F_{2^k}Q_8$, Internat. J. Algebra Comput., 19(2) (2009), 283-286.
• L. Creedon and J. Gildea, The structure of the unit group of the group algebra $\Bbb F_{2^k}D_8$, Canad. Math. Bull., 54(2) (2011), 237-243.
• E. T. Hill, The annihilator of radical powers in the modular group ring of a p-group, Proc. Amer. Math. Soc., 25 (1970), 811-815.
• J.-P. Serre, Bases normales autoduales et groupes unitaires en caracteristique 2, Transform. Groups, 19(2) (2014), 643-698.