Geodesic
Principal blocks and $p$-radical groups
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 431-444
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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.
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.
DOI : 10.1007/s10587-016-0266-x
Classification : 20C05, 20C20
Keywords: principal block; $p$-radical group; $p$-radical block 
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Hu, Xiaohan; Zeng, Jiwen. Principal blocks and $p$-radical groups. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 431-444. doi: 10.1007/s10587-016-0266-x

[1] Feit, W.: The Representation Theory of Finite Groups. North-Holland Mathematical Library 25 North-Holland, Amsterdam (1982). | MR | Zbl

[2] Fong, P.: Solvable groups and modular representation theory. Trans. Am. Math. Soc. 103 (1962), 484-494. | DOI | MR | Zbl

[3] Fong, P., Gaschütz, W.: A note on the modular representations of solvable groups. J. Reine Angew. Math. 208 (1961), 73-78. | MR | Zbl

[4] Gorenstein, D.: Finite Groups. Chelsea Publishing Company New York (1980). | MR | Zbl

[5] Hida, A.: On $p$-radical blocks of finite groups. Proc. Am. Math. Soc. 114 (1992), 37-38. | MR | Zbl

[6] Huppert, B., Blackburn, N.: Finite Groups II. Grundlehren der Mathematischen Wissenschaften 242 Springer, Berlin (1982). | MR | Zbl

[7] Huppert, B., Blackburn, N.: Finite Groups III. Grundlehren der Mathematischen Wissenschaften 243 Springer, Berlin (1982). | DOI | MR | Zbl

[8] Karpilovsky, G.: The Jacobson Radical of Group Algebras. North-Holland Mathematics Studies 135, Notas de Matemática 115 North-Holland, Amsterdam (1987). | MR | Zbl

[9] Knörr, R.: On the vertices of irreducible modules. Ann. Math. 110 (1979), 487-499. | DOI | MR | Zbl

[10] Knörr, R.: Semisimplicity, induction, and restriction for modular representations of finite groups. J. Algebra 48 (1977), 347-367. | DOI | MR | Zbl

[11] Knörr, R.: Blocks, vertices and normal subgroups. Math. Z. 148 (1976), 53-60. | DOI | MR | Zbl

[12] Koshitani, S.: A remark on $p$-radical groups. J. Algebra 134 (1990), 491-496. | DOI | MR | Zbl

[13] Laradji, A.: A characterization of $p$-radical groups. J. Algebra 188 (1997), 686-691. | DOI | MR | Zbl

[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 | Zbl

[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

[16] Nagao, H., Tsushima, Y.: Representations of Finite Groups. Academic Press Boston (1989). | MR | Zbl

[17] Ninomiya, Y.: Structure of $p$-solvable groups with three $p$-regular classes. II. Math. J. Okayama Univ. 35 (1993), 29-34. | MR | Zbl

[18] Ninomiya, Y.: Structure of $p$-solvable groups with three $p$-regular classes. Can. J. Math. 43 (1991), 559-579. | DOI | MR | Zbl

[19] Okuyama, T.: $p$-radical groups are $p$-solvable. Osaka J. Math. 23 (1986), 467-469. | MR | Zbl

[20] Okuyama, T.: Module correspondence in finite groups. Hokkaido Math. J. 10 (1981), 299-318. | DOI | MR | Zbl

[21] Passman, D.: Permutation Groups. Benjamin New York (1968). | MR | Zbl

[22] Saksonov, A. I.: On the decomposition of a permutation group over a characteristic field. Sov. Math., Dokl. 12 (1971), 786-790. | MR | Zbl

[23] Tsushima, Y.: On $p$-radical groups. J. Algebra 103 (1986), 80-86. | DOI | MR | Zbl

[24] Tsushima, Y.: On the second reduction theorem of P. Fong. Kumamoto J. Sci., Math. 13 (1978), 6-14. | MR | Zbl

[25] Wallace, D. A. R.: On the commutativity of the radical of a group algebra. Proc. Glasg. Math. Assoc. 7 (1965), 1-8. | DOI | MR | Zbl

[26] Wallace, D. A. R.: Group algebras with radicals of square zero. Proc. Glasg. Math. Assoc. 5 (1962), 158-159. | DOI | MR | Zbl

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