The Weights of Irreducible SL 3(q)-Modules in the Defining Characteristic

This is a final step in solving the problem of recognition of the simple groups L ₃(p ^k) by element orders. It is proven that when L ₃(p ^k) acts on an elementary abelian p-group, there always appears an element of new order. A model is proposed for constructing the absolutely irreducible p-modular representations of L ₃(p ^k) in polynomial spaces.