Principal blocks and p -radical groups

Xiaohan Hu; Jiwen Zeng

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 2, page 431-444
  • ISSN: 0011-4642

Abstract

top
Let G be a finite group and k a field of characteristic p > 0 . In this paper, we obtain several equivalent conditions to determine whether the principal block B 0 of a finite p -solvable group G is p -radical, which means that B 0 has the property that e 0 ( k P ) G is semisimple as a k G -module, where P is a Sylow p -subgroup of G , k P is the trivial k P -module, ( k P ) G is the induced module, and e 0 is the block idempotent of B 0 . We also give the complete classification of a finite p -solvable group G which has not more than three simple B 0 -modules where B 0 is p -radical.

How to cite

top

Hu, Xiaohan, and Zeng, Jiwen. "Principal blocks and $p$-radical groups." Czechoslovak Mathematical Journal 66.2 (2016): 431-444. <http://eudml.org/doc/280104>.

@article{Hu2016,
abstract = {Let $G$ be a finite group and $k$ a field of characteristic $p > 0$. In this paper, we obtain several equivalent conditions to determine whether the principal block $B_\{0\}$ of a finite $p$-solvable group $G$ is $p$-radical, which means that $B_\{0\}$ has the property that $e_\{0\} (k_P)^G $ is semisimple as a $kG$-module, where $P$ is a Sylow $p$-subgroup of $G$, $k_\{P\}$ is the trivial $kP$-module, $(k_\{P\})^\{G\}$ is the induced module, and $e_\{0\}$ is the block idempotent of $B_\{0\}$. We also give the complete classification of a finite $p$-solvable group $G$ which has not more than three simple $B_\{0\}$-modules where $B_0$ is $p$-radical.},
author = {Hu, Xiaohan, Zeng, Jiwen},
journal = {Czechoslovak Mathematical Journal},
keywords = {principal block; $p$-radical group; $p$-radical block},
language = {eng},
number = {2},
pages = {431-444},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Principal blocks and $p$-radical groups},
url = {http://eudml.org/doc/280104},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Hu, Xiaohan
AU - Zeng, Jiwen
TI - Principal blocks and $p$-radical groups
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 2
SP - 431
EP - 444
AB - Let $G$ be a finite group and $k$ a field of characteristic $p > 0$. In this paper, we obtain several equivalent conditions to determine whether the principal block $B_{0}$ of a finite $p$-solvable group $G$ is $p$-radical, which means that $B_{0}$ has the property that $e_{0} (k_P)^G $ is semisimple as a $kG$-module, where $P$ is a Sylow $p$-subgroup of $G$, $k_{P}$ is the trivial $kP$-module, $(k_{P})^{G}$ is the induced module, and $e_{0}$ is the block idempotent of $B_{0}$. We also give the complete classification of a finite $p$-solvable group $G$ which has not more than three simple $B_{0}$-modules where $B_0$ is $p$-radical.
LA - eng
KW - principal block; $p$-radical group; $p$-radical block
UR - http://eudml.org/doc/280104
ER -

References

top
  1. Feit, W., The Representation Theory of Finite Groups, North-Holland Mathematical Library 25 North-Holland, Amsterdam (1982). (1982) Zbl0493.20007MR0661045
  2. Fong, P., 10.1090/S0002-9947-1962-0139667-5, Trans. Am. Math. Soc. 103 (1962), 484-494. (1962) Zbl0105.25603MR0139667DOI10.1090/S0002-9947-1962-0139667-5
  3. Fong, P., Gaschütz, W., A note on the modular representations of solvable groups, J. Reine Angew. Math. 208 (1961), 73-78. (1961) Zbl0100.25801MR0138690
  4. Gorenstein, D., Finite Groups, Chelsea Publishing Company New York (1980). (1980) Zbl0463.20012MR0569209
  5. Hida, A., On p -radical blocks of finite groups, Proc. Am. Math. Soc. 114 (1992), 37-38. (1992) Zbl0744.20008MR1069688
  6. Huppert, B., Blackburn, N., Finite Groups II, Grundlehren der Mathematischen Wissenschaften 242 Springer, Berlin (1982). (1982) Zbl0477.20001MR0650245
  7. Huppert, B., Blackburn, N., 10.1007/978-3-642-67997-1_1, Grundlehren der Mathematischen Wissenschaften 243 Springer, Berlin (1982). (1982) Zbl0514.20002MR0662826DOI10.1007/978-3-642-67997-1_1
  8. Karpilovsky, G., The Jacobson Radical of Group Algebras, North-Holland Mathematics Studies 135, Notas de Matemática 115 North-Holland, Amsterdam (1987). (1987) Zbl0618.16001MR0886889
  9. Knörr, R., 10.2307/1971234, Ann. Math. 110 (1979), 487-499. (1979) Zbl0388.20004MR0554380DOI10.2307/1971234
  10. Knörr, R., 10.1016/0021-8693(77)90313-1, J. Algebra 48 (1977), 347-367. (1977) Zbl0412.20007MR0466289DOI10.1016/0021-8693(77)90313-1
  11. Knörr, R., 10.1007/BF01187868, Math. Z. 148 (1976), 53-60. (1976) Zbl0308.20013MR0401897DOI10.1007/BF01187868
  12. Koshitani, S., 10.1016/0021-8693(90)90063-T, J. Algebra 134 (1990), 491-496. (1990) Zbl0713.20004MR1074339DOI10.1016/0021-8693(90)90063-T
  13. Laradji, A., 10.1006/jabr.1996.6830, J. Algebra 188 (1997), 686-691. (1997) Zbl0876.20001MR1435380DOI10.1006/jabr.1996.6830
  14. Morita, K., On group rings over a modular field which possess radicals expressible as principal ideals, Sci. Rep. Tokyo Bunrika Daikagu, Sect. A 4 (1951), 177-194. (1951) Zbl0053.35002MR0049909
  15. Motose, K., Ninomiya, Y., On the subgroups H of a group G such that 𝒥 ( K H ) K G ⊃ 𝒥 ( K G ) , Math. J. Okayama Univ. 17 (1975), 171-176. (1975) MR0376837
  16. Nagao, H., Tsushima, Y., Representations of Finite Groups, Academic Press Boston (1989). (1989) Zbl0673.20002MR0998775
  17. Ninomiya, Y., Structure of p -solvable groups with three p -regular classes. II, Math. J. Okayama Univ. 35 (1993), 29-34. (1993) Zbl0826.20012MR1329910
  18. Ninomiya, Y., 10.4153/CJM-1991-034-2, Can. J. Math. 43 (1991), 559-579. (1991) Zbl0738.20012MR1118010DOI10.4153/CJM-1991-034-2
  19. Okuyama, T., p -radical groups are p -solvable, Osaka J. Math. 23 (1986), 467-469. (1986) Zbl0611.20006MR0856900
  20. Okuyama, T., 10.14492/hokmj/1381758078, Hokkaido Math. J. 10 (1981), 299-318. (1981) Zbl0488.20013MR0634165DOI10.14492/hokmj/1381758078
  21. Passman, D., Permutation Groups, Benjamin New York (1968). (1968) Zbl0179.04405MR0237627
  22. Saksonov, A. I., On the decomposition of a permutation group over a characteristic field, Sov. Math., Dokl. 12 (1971), 786-790. (1971) Zbl0235.20011MR0318281
  23. Tsushima, Y., 10.1016/0021-8693(86)90169-9, J. Algebra 103 (1986), 80-86. (1986) Zbl0597.16011MR0860689DOI10.1016/0021-8693(86)90169-9
  24. Tsushima, Y., On the second reduction theorem of P. Fong, Kumamoto J. Sci., Math. 13 (1978), 6-14. (1978) Zbl0385.20004MR0491921
  25. Wallace, D. A. R., 10.1017/S2040618500035061, Proc. Glasg. Math. Assoc. 7 (1965), 1-8. (1965) Zbl0127.01402MR0178074DOI10.1017/S2040618500035061
  26. Wallace, D. A. R., 10.1017/S2040618500034523, Proc. Glasg. Math. Assoc. 5 (1962), 158-159. (1962) Zbl0105.02402MR0140568DOI10.1017/S2040618500034523

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.