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ON UNITARY SUBGROUPS OF GROUP ALGEBRAS

Year 2021, , 187 - 198, 05.01.2021
https://doi.org/10.24330/ieja.852199

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.

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.
Year 2021, , 187 - 198, 05.01.2021
https://doi.org/10.24330/ieja.852199

Abstract

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.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Zsolt Adam Balogh This is me

Publication Date January 5, 2021
Published in Issue Year 2021

Cite

APA Balogh, Z. A. (2021). ON UNITARY SUBGROUPS OF GROUP ALGEBRAS. International Electronic Journal of Algebra, 29(29), 187-198. https://doi.org/10.24330/ieja.852199
AMA Balogh ZA. ON UNITARY SUBGROUPS OF GROUP ALGEBRAS. IEJA. January 2021;29(29):187-198. doi:10.24330/ieja.852199
Chicago Balogh, Zsolt Adam. “ON UNITARY SUBGROUPS OF GROUP ALGEBRAS”. International Electronic Journal of Algebra 29, no. 29 (January 2021): 187-98. https://doi.org/10.24330/ieja.852199.
EndNote Balogh ZA (January 1, 2021) ON UNITARY SUBGROUPS OF GROUP ALGEBRAS. International Electronic Journal of Algebra 29 29 187–198.
IEEE Z. A. Balogh, “ON UNITARY SUBGROUPS OF GROUP ALGEBRAS”, IEJA, vol. 29, no. 29, pp. 187–198, 2021, doi: 10.24330/ieja.852199.
ISNAD Balogh, Zsolt Adam. “ON UNITARY SUBGROUPS OF GROUP ALGEBRAS”. International Electronic Journal of Algebra 29/29 (January 2021), 187-198. https://doi.org/10.24330/ieja.852199.
JAMA Balogh ZA. ON UNITARY SUBGROUPS OF GROUP ALGEBRAS. IEJA. 2021;29:187–198.
MLA Balogh, Zsolt Adam. “ON UNITARY SUBGROUPS OF GROUP ALGEBRAS”. International Electronic Journal of Algebra, vol. 29, no. 29, 2021, pp. 187-98, doi:10.24330/ieja.852199.
Vancouver Balogh ZA. ON UNITARY SUBGROUPS OF GROUP ALGEBRAS. IEJA. 2021;29(29):187-98.