Serial group rings of finite groups. Sporadic simple groups and Suzuki groups
type: Item
instance of: scholarly article
title: Serial group rings of finite groups. Sporadic simple groups and Suzuki groups
zbMATH Open document ID: 1362.20005
author: A. V. Kukharev
author: G. E. Puninskij
published in: Journal of Mathematical Sciences (New York)
publication date: 2017-01-24T00:00:00Z
review text: 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 \(BM\) …
reviewed by: János Kurdics
Mathematics Subject Classification ID: 20C34
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20D08
zbMATH DE Number: 6676943
zbMATH Keywords: finite group
zbMATH Keywords: group algebra
zbMATH Keywords: simple
zbMATH Keywords: sporadic
zbMATH Keywords: Suzuki
zbMATH Keywords: uniserial
zbMATH Keywords: serial
zbMATH Keywords: serial group ring
zbMATH Keywords: p-solvable
zbMATH Keywords: block
zbMATH Keywords: defect
zbMATH Keywords: Brauer tree
describes a project that uses: ATLAS Group Representations
describes a project that uses: GAP
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: s10958-016-3126-1
OpenAlex ID: W2541244046
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DOI: 10.1007/S10958-016-3126-1
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description: scientific article; zbMATH DE number 6676943