Previous |  Up |  Next

Article

MSC: 20C05, 20C20
Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
principal block; $p$-radical group; $p$-radical block
Summary:
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.
References:
[1] Feit, W.: The Representation Theory of Finite Groups. North-Holland Mathematical Library 25 North-Holland, Amsterdam (1982). MR 0661045 | Zbl 0493.20007
[2] Fong, P.: Solvable groups and modular representation theory. Trans. Am. Math. Soc. 103 (1962), 484-494. DOI 10.1090/S0002-9947-1962-0139667-5 | MR 0139667 | Zbl 0105.25603
[3] Fong, P., Gaschütz, W.: A note on the modular representations of solvable groups. J. Reine Angew. Math. 208 (1961), 73-78. MR 0138690 | Zbl 0100.25801
[4] Gorenstein, D.: Finite Groups. Chelsea Publishing Company New York (1980). MR 0569209 | Zbl 0463.20012
[5] Hida, A.: On $p$-radical blocks of finite groups. Proc. Am. Math. Soc. 114 (1992), 37-38. MR 1069688 | Zbl 0744.20008
[6] Huppert, B., Blackburn, N.: Finite Groups II. Grundlehren der Mathematischen Wissenschaften 242 Springer, Berlin (1982). MR 0650245 | Zbl 0477.20001
[7] Huppert, B., Blackburn, N.: Finite Groups III. Grundlehren der Mathematischen Wissenschaften 243 Springer, Berlin (1982). DOI 10.1007/978-3-642-67997-1_1 | MR 0662826 | Zbl 0514.20002
[8] Karpilovsky, G.: The Jacobson Radical of Group Algebras. North-Holland Mathematics Studies 135, Notas de Matemática 115 North-Holland, Amsterdam (1987). MR 0886889 | Zbl 0618.16001
[9] Knörr, R.: On the vertices of irreducible modules. Ann. Math. 110 (1979), 487-499. DOI 10.2307/1971234 | MR 0554380 | Zbl 0388.20004
[10] Knörr, R.: Semisimplicity, induction, and restriction for modular representations of finite groups. J. Algebra 48 (1977), 347-367. DOI 10.1016/0021-8693(77)90313-1 | MR 0466289 | Zbl 0412.20007
[11] Knörr, R.: Blocks, vertices and normal subgroups. Math. Z. 148 (1976), 53-60. DOI 10.1007/BF01187868 | MR 0401897 | Zbl 0308.20013
[12] Koshitani, S.: A remark on $p$-radical groups. J. Algebra 134 (1990), 491-496. DOI 10.1016/0021-8693(90)90063-T | MR 1074339 | Zbl 0713.20004
[13] Laradji, A.: A characterization of $p$-radical groups. J. Algebra 188 (1997), 686-691. DOI 10.1006/jabr.1996.6830 | MR 1435380 | Zbl 0876.20001
[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. MR 0049909 | Zbl 0053.35002
[15] Motose, K., Ninomiya, Y.: On the subgroups $H$ of a group $G$ such that $\mathcal{J}(KH)KG\supset$ $\mathcal{J}(KG)$. Math. J. Okayama Univ. 17 (1975), 171-176. MR 0376837
[16] Nagao, H., Tsushima, Y.: Representations of Finite Groups. Academic Press Boston (1989). MR 0998775 | Zbl 0673.20002
[17] Ninomiya, Y.: Structure of $p$-solvable groups with three $p$-regular classes. II. Math. J. Okayama Univ. 35 (1993), 29-34. MR 1329910 | Zbl 0826.20012
[18] Ninomiya, Y.: Structure of $p$-solvable groups with three $p$-regular classes. Can. J. Math. 43 (1991), 559-579. DOI 10.4153/CJM-1991-034-2 | MR 1118010 | Zbl 0738.20012
[19] Okuyama, T.: $p$-radical groups are $p$-solvable. Osaka J. Math. 23 (1986), 467-469. MR 0856900 | Zbl 0611.20006
[20] Okuyama, T.: Module correspondence in finite groups. Hokkaido Math. J. 10 (1981), 299-318. DOI 10.14492/hokmj/1381758078 | MR 0634165 | Zbl 0488.20013
[21] Passman, D.: Permutation Groups. Benjamin New York (1968). MR 0237627 | Zbl 0179.04405
[22] Saksonov, A. I.: On the decomposition of a permutation group over a characteristic field. Sov. Math., Dokl. 12 (1971), 786-790. MR 0318281 | Zbl 0235.20011
[23] Tsushima, Y.: On $p$-radical groups. J. Algebra 103 (1986), 80-86. DOI 10.1016/0021-8693(86)90169-9 | MR 0860689 | Zbl 0597.16011
[24] Tsushima, Y.: On the second reduction theorem of P. Fong. Kumamoto J. Sci., Math. 13 (1978), 6-14. MR 0491921 | Zbl 0385.20004
[25] Wallace, D. A. R.: On the commutativity of the radical of a group algebra. Proc. Glasg. Math. Assoc. 7 (1965), 1-8. DOI 10.1017/S2040618500035061 | MR 0178074 | Zbl 0127.01402
[26] Wallace, D. A. R.: Group algebras with radicals of square zero. Proc. Glasg. Math. Assoc. 5 (1962), 158-159. DOI 10.1017/S2040618500034523 | MR 0140568 | Zbl 0105.02402
Partner of
EuDML logo