Finite groups whose first Cartan invariants over a field of characteristic two are two

Abstract

We determine all finite groups whose first Cartan invariants over a field of characteristic two are two, by using the classification of finite simple groups.

The Projective Cover of the Trivial Representation for a Finite Group of Lie Type in Defining Characteristic

“…Then it follows from results of Okuyama [34] and Erdmann [8] that LL(P (k G )) = 3 if and only if G = SL 3 (2), see [22,Theorem 3.3]; and by [22, Theorem 1.2] we always have LL(P (k G )) = 4. Moreover, by [23,Lemma 4.5] we have c 11 (G) = 2 if and only if G = SL 3 (2). Hence in conclusion for this class of groups we have LL(P (k G )) ≥ 5 if and only if c 11 (G) ≥ 3 if and only if G ∼ = SL 3 (2).…”