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 over a field of characteristic ; and let be the group of units of *FG*. We prove that is solvable if and only if (i) every absolutely irreducible representation of at characteristic is of degree one or two and (ii) if any such representation is of degree two, then it is definable in and or . This result is translated into intrinsic group-theoretic and field-theoretic conditions on and , respectively. Namely, if is the maximum normal -subgroup of and , then (i) is abelian, or (ii) and is a -group with exactly normal subgroups of index 8 that do not contain , or (iii) and is the extension of an elementary abelian -group by an automorphism which inverts every element.

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

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DOI:
https://doi.org/10.1090/S0002-9947-1971-0276371-2

Keywords:
Group algebra,
unit group,
radical,
separable algebra,
solvability,
irreducible representations

Article copyright:
© Copyright 1971
American Mathematical Society