Preradicals Over Some Group Algebras

For a field \(\varvec{K}\) and a finite group \(\varvec{G}\), we study the lattice of preradicals over the group algebra \(\varvec{KG}\), denoted by \(\varvec{KG}\)-\(\varvec{pr}\). We show that if \(\varvec{KG}\) is a semisimple algebra, then \(\varvec{KG}\)-\(\varvec{pr}\) is completely described, and we establish conditions for counting the number of its atoms in some specific cases. If \(\varvec{KG}\) is an algebra of finite representation type, but not a semisimple one, we completely describe \(\varvec{KG}\)-\(\varvec{pr}\) when the characteristic of \(\varvec{K}\) is a prime \(\varvec{p}\) and \(\varvec{G}\) is a cyclic \(\varvec{p}\)-group. For group algebras of infinite representation type, we show that the lattices of preradicals over two representative families of such algebras are not sets (in which case, we say the algebras are \(\varvec{\mathfrak {p}}\)-large). Besides, we provide new examples of \(\varvec{\mathfrak {p}}\)-large algebras. Finally, we prove the main theorem of this paper which characterizes the representation type of group algebras \(\varvec{KG}\) in terms of their lattice of preradicals.