Serial Group Rings of Finite Simple Groups of Lie Type

Abstract

Suppose that F is a field whose characteristic p divides the order of a finite group G. It is shown that if G is one of the groups 3D4(q), E6(q), 2E6(q), E7(q), E8(q), F4(q), 2F4(q), or 2G2(q), then the group ring FG is not serial. If G = G2(q2), then the ring FG is serial if and only if either p > 2 divides q2 − 1, or p = 7 divides $$ {q}^2+\sqrt{3q}+1 $$ but 49 does not divide this number.