in this paper, we introduce $p$-semilinear transformations for linear algebras over a field ${bf f}$ of positive characteristic $p$, discuss initially the elementary properties of $p$-semilinear transformations, make use of it to give some characterizations of linear algebras over a field ${bf f}$ of positive characteristic $p$. moreover, we find a one-to-one correspondence between $p$-semiline...

In this paper, we introduce $p$-semilinear transformations for linear algebras over a field ${bf F}$ of positive characteristic $p$, discuss initially the elementary properties of $p$-semilinear transformations, make use of it to give some characterizations of linear algebras over a field ${bf F}$ of positive characteristic $p$. Moreover, we find a one-to-one correspondence between $p$-semiline...

Notes for a course, held in the Biebrza Marshes, Poland, May 6–10, 2010, on Frobenius categories of finite groups of Lie type.

Let G be a group and D be a subgroup of G. In this paper we are interested in the description of the lattice L(D; G) = fH j D H Gg of subgroups of G which contain D. We call subgroups H 2 L(D; G) subgroups intermediate between D and G. We are primarily interested in the parametrization of intermediate subgroups and their behaviour with respect to set-theoretic and group-theoretic operations lik...

The paper is devoted to a detailed study of some remarkable semisimple elements of (extended) Chevalley groups that are diagonalizable over the ground field — the weight elements. These are the conjugates of certain semisimple elements hω(ε) of extended Chevalley groupsG = G(Φ, K), where ω is a weight of the dual root system Φ∨ and ε ∈ K∗. In the adjoint case the hω(ε)’s were defined by Chevall...