Serial group rings of finite groups. Sporadic simple groups and Suzuki groups
Let \(G\) be a finite group, \(F\) a field of prime characteristic \(p\), \(FG\) the group algebra. The authors, continuing their research in [J. Math. Sci., New York 202, No. 3, 422--433 (2014; Zbl 1311.20001)] and [Algebra Discrete Math. 16, No. 2, 201--216 (2013; Zbl 1311.20002)], study the question: when the group ring \(FG\) is serial? A module is said to be uniserial if its submodules form a chain, and serial if it is the direct sum of uniserial modules. A ring is serial if it is serial as a right and as a left module over itself. By \textit{D. G. Higman}'s [Duke Math. J. 21, 377--381 (1954; Zbl 0055.25503)] classical result, if \(FG\) is serial then any Sylow \(p\)-subgroup is cyclic. It is also well-known that for algebraically closed \(F\), \(FG\) is serial if and only if the Brauer tree of each block of \(G\) is serial with exceptional vertex (if it exists) in the centre. Moreover, by \textit{K. Morita} [Sci. Rep. Tokyo Bunrika Daigaku, Sect. A 4, 177--194 (1951; Zbl 0053.35002)], if \(G\) is \(p\)-solvable with a cyclic Sylow \(p\)-subgroup then \(FG\) is serial. Let \(G\) be not \(p\)-solvable with a cyclic Sylow \(p\)-subgroup. Then there exists the smallest normal subgroup \(K\) properly containing \(O_{p'}\), and the factor group \(H = K/O_{p'}\) is simple non-abelian. The authors conjecture the following: The following statements are equivalent for \(G\) possessing a cyclic Sylow \(p\)-subgroup: (1) \(FG\) is serial; (2) either \(G\) is \(p\)-solvable or the ring \(FH\) is serial. The present paper determines the sporadic and Suzuki simple groups \(G\) and primes \(p\) such that the modular group algebra \(FG\) is serial, namely these are (1) \(G=M_{11}\), \(p=5\); (2) \(G=J_1\), \(p=3\); or \(G=Sz(q)\), \(q=2^{2n+1}\), \(r = 2^{n+1}\), \(n\geq1\), (3) \(p\) divides \(q-1\) or (4) \(p=5\) divides \(q+r+1\) but \(5^2\) does not divide \(q+r+1\). In most cases, the information on the Brauer tree is known, the remaining cases of \(Fi'_{24}\) and \(BM\) are handled in the paper, summing up in a table the structure of the Brauer trees in all cases. The following two problems remain open: (1) Checking seriality for all remaining finite simple groups; and (2) Proving sufficiency of the conjecture.
scientific article; zbMATH DE number 6676943
scientific article