Computing the order of the nilpotent residual of a finite group from knowledge of its group algebra
The authors show that the group-algebra \(FG\) of a finite group over an algebraically closed field determines the order of \(K_ \infty(G)\), i.e. the order of the last term of the lower central series. This extends a result of I. M. Isaacs, who showed that for \((\text{char}(K),| G|) = 1\) the isomorphism \(FG \simeq FH\) implies that \(H\) is nilpotent provided \(G\) is nilpotent (just take the case where \(K_ \infty(G) = \langle 1\rangle\)). As crucial point in the arguments of the authors they prove: A finite group \(G\) is \(p\)-nilpotent if and only if the group-algebra \(FG\) over an algebraically closed field contains a block of dimension \(| G|_ p\). In order to derive the final result, the authors prove -- using modular representation theory: Assume that \(P\) is a minimal right ideal of \(FG\) for a finite group \(G\) and an algebraically closed field \(F\) of characteristic \(p > 0\), such that \(\text{Dim}_ F(P/\text{rad}(P)) = 1\) (the authors mean -- without saying this -- that \(P\) is a projective right ideal). Let \(FG = P \oplus Q\) as right modules and put \(P_ 0 := \bigcap^ \infty_{i = 0} P \cdot Q^ n\). Then 1. \(| G/O^ p(G)| = \text{Dim}_ F(P/P_ 0)\), and 2. if \(P/R\) is the largest commutative quotient algebra of \(P/P_ 0\), then \(| G/G'|_ p = \text{Dim}_ F(P/R)\).
scientific article; zbMATH DE number 571905