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.