Characterizing projective general unitary groups PGU 3 ( q 2 ) by their complex group algebras

Farrokh Shirjian; Ali Iranmanesh

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 3, page 819-826
  • ISSN: 0011-4642

Abstract

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Let G be a finite group. Let X 1 ( G ) be the first column of the ordinary character table of G . We will show that if X 1 ( G ) = X 1 ( PGU 3 ( q 2 ) ) , then G ≅ PGU 3 ( q 2 ) . As a consequence, we show that the projective general unitary groups PGU 3 ( q 2 ) are uniquely determined by the structure of their complex group algebras.

How to cite

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Shirjian, Farrokh, and Iranmanesh, Ali. "Characterizing projective general unitary groups ${\rm PGU}_3(q^2)$ by their complex group algebras." Czechoslovak Mathematical Journal 67.3 (2017): 819-826. <http://eudml.org/doc/294105>.

@article{Shirjian2017,
abstract = {Let $G$ be a finite group. Let $X_1(G)$ be the first column of the ordinary character table of $G$. We will show that if $X_1(G)=X_1(\{\rm PGU\}_3(q^2))$, then $G \cong \{\rm PGU\}_3(q^2)$. As a consequence, we show that the projective general unitary groups $\{\rm PGU\}_3(q^2)$ are uniquely determined by the structure of their complex group algebras.},
author = {Shirjian, Farrokh, Iranmanesh, Ali},
journal = {Czechoslovak Mathematical Journal},
keywords = {character degree; complex group algebra; projective general unitary group},
language = {eng},
number = {3},
pages = {819-826},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Characterizing projective general unitary groups $\{\rm PGU\}_3(q^2)$ by their complex group algebras},
url = {http://eudml.org/doc/294105},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Shirjian, Farrokh
AU - Iranmanesh, Ali
TI - Characterizing projective general unitary groups ${\rm PGU}_3(q^2)$ by their complex group algebras
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 3
SP - 819
EP - 826
AB - Let $G$ be a finite group. Let $X_1(G)$ be the first column of the ordinary character table of $G$. We will show that if $X_1(G)=X_1({\rm PGU}_3(q^2))$, then $G \cong {\rm PGU}_3(q^2)$. As a consequence, we show that the projective general unitary groups ${\rm PGU}_3(q^2)$ are uniquely determined by the structure of their complex group algebras.
LA - eng
KW - character degree; complex group algebra; projective general unitary group
UR - http://eudml.org/doc/294105
ER -

References

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  1. Berkovich, Y. G., Zhmud', E. M., Characters of Finite Groups. Part 1, Translations of Mathematical Monographs 172, American Mathematical Society, Providence (1998). (1998) Zbl0934.20008MR1486039
  2. Bessenrodt, C., Nguyen, H. N., Olsson, J. B., Tong-Viet, H. P., 10.2140/ant.2015.9.601, Algebra Number Theory 9 (2015), 601-628. (2015) Zbl1321.20011MR3340546DOI10.2140/ant.2015.9.601
  3. Brauer, R., Representations of finite groups, Lect. Modern Math. 1 (1963), 133-175. (1963) Zbl0124.26504MR0178056
  4. Carter, R. W., Finite Groups of Lie Type. Conjugacy Classes and Complex Characters, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Chichester (1993). (1993) Zbl0567.20023MR1266626
  5. Dolfi, S., Navarro, G., Tiep, P. H., 10.1007/s00209-007-0247-8, Math. Z. 259 (2008), 755-774. (2008) Zbl1149.20006MR2403740DOI10.1007/s00209-007-0247-8
  6. Gorenstein, D., Lyons, R., Solomon, R., The Classification of the Finite Simple Groups. Part I. Chapter A: Almost Simple K -Groups, Mathematical Surveys and Monographs 40, American Mathematical Society, Providence (1998). (1998) Zbl0890.20012MR1490581
  7. Heydari, S., Ahanjideh, N., 10.2298/PIM150111017H, Publ. Inst. Math., Nouv. Sér. 99 (2016), 257-264. (2016) Zbl06749629MR3524049DOI10.2298/PIM150111017H
  8. Huppert, B., Some simple groups, which are determined by the set of their character degrees III, Preprint. MR1804317
  9. Kimmerle, W., Group rings of finite simple groups, Resen. Inst. Mat. Estat. Univ. São Paulo 5 (2002), 261-278. (2002) Zbl1047.20007MR2015338
  10. Lewis, M. L., 10.1515/jgth.2001.023, J. Group Theory 4 (2001), 255-275. (2001) Zbl0998.20009MR1839998DOI10.1515/jgth.2001.023
  11. Lübeck, F., Data for finite groups of Lie type and related algebraic groups, Available at http://www.math.rwth-aachen.de/Frank.Luebeck/chev/index.html. 
  12. Malle, G., Testerman, D., 10.1017/CBO9780511994777, Cambridge Studies in Advanced Mathematics 133, Cambridge University Press, Cambridge (2011). (2011) Zbl1256.20045MR2850737DOI10.1017/CBO9780511994777
  13. Meng, Q., Zeng, J., 10.1142/S1005386713000060, Algebra Colloq. 20 (2013), 75-80. (2013) Zbl1280.20009MR3020718DOI10.1142/S1005386713000060
  14. Nagl, M., Charakterisierung der Symmetrischen Gruppen durch ihre komplexe Gruppenalgebra, Stuttgarter Mathematische Berichte (2011), German Available at http://www.mathematik.uni-stuttgart.de/preprints/downloads/2011/2011-007.pdf. (2011) 
  15. Nguyen, H. N., 10.1007/s11856-012-0142-9, Isr. J. Math. 195 (2013), 973-998. (2013) Zbl1294.20004MR3096579DOI10.1007/s11856-012-0142-9
  16. Nguyen, H. N., Tong-Viet, H. P., 10.1007/s10468-012-9400-0, Algebr. Represent. Theory 17 (2014), 305-320. (2014) Zbl1303.20001MR3160726DOI10.1007/s10468-012-9400-0
  17. Shirjian, F., Iranmanesh, A., 10.1080/00927872.2017.1324868, To appear in Commun. Algebra. DOI10.1080/00927872.2017.1324868
  18. Simpson, W. A., Frame, J. S., 10.4153/CJM-1973-049-7, Can. J. Math. 25 (1973), 486-494. (1973) Zbl0264.20010MR0335618DOI10.4153/CJM-1973-049-7
  19. Tong-Viet, H. P., 10.1016/j.jalgebra.2010.11.018, J. Algebra 334 (2011), 275-284. (2011) Zbl1246.20007MR2787664DOI10.1016/j.jalgebra.2010.11.018
  20. Tong-Viet, H. P., 10.1007/s10468-010-9247-1, Algebr. Represent. Theory 15 (2012), 379-389. (2012) Zbl1252.20005MR2892513DOI10.1007/s10468-010-9247-1
  21. Tong-Viet, H. P., 10.1016/j.jalgebra.2012.02.011, J. Algebra 357 (2012), 61-68. (2012) Zbl1259.20008MR2905242DOI10.1016/j.jalgebra.2012.02.011
  22. Tong-Viet, H. P., 10.1007/s00605-011-0301-9, Monatsh. Math. 166 (2012), 559-577. (2012) Zbl1255.20006MR2925155DOI10.1007/s00605-011-0301-9
  23. Wakefield, T. P., 10.1080/00927870802625661, Commun. Algebra 37 (2009), 2887-2906. (2009) Zbl1185.20014MR2543522DOI10.1080/00927870802625661
  24. Zsigmondy, K., 10.1007/BF01692444, Monatsh. Math. Phys. 3 (1892), 265-284 German 9999JFM99999 24.0176.02. (1892) MR1546236DOI10.1007/BF01692444

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