On the solvability of unit groups of group algebras

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 $(|L| - [L:L’])/4$ 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 $U$.