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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the solvability of unit groups of group algebras


Author: J. M. Bateman
Journal: Trans. Amer. Math. Soc. 157 (1971), 73-86
MSC: Primary 20.80
MathSciNet review: 0276371
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Abstract: Let FG be the group algebra of a finite group $ G$ over a field $ F$ of characteristic $ p \geqq 0$; and let $ U$ be the group of units of FG. We prove that $ U$ is solvable if and only if (i) every absolutely irreducible representation of $ G$ at characteristic $ p$ is of degree one or two and (ii) if any such representation is of degree two, then it is definable in $ F$ and $ F = GF(2)$ or $ GF(3)$. This result is translated into intrinsic group-theoretic and field-theoretic conditions on $ G$ and $ F$, respectively. Namely, if $ {O_p}(G)$ is the maximum normal $ p$-subgroup of $ G$ and $ L = G/{O_p}(G)$, then (i) $ L$ is abelian, or (ii) $ F = GF(3)$ and $ L$ is a $ 2$-group with exactly $ (\vert L\vert - [L:L'])/4$ normal subgroups of index 8 that do not contain $ L'$, or (iii) $ F = GF(2)$ and $ L$ is the extension of an elementary abelian $ 3$-group by an automorphism which inverts every element.

Conditions are found for the nilpotency, supersolvability, and $ p$-solvability of $ U$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0276371-2
PII: S 0002-9947(1971)0276371-2
Keywords: Group algebra, unit group, radical, separable algebra, solvability, irreducible representations
Article copyright: © Copyright 1971 American Mathematical Society