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Let   ${\textbf{G}}$   be a connected reductive algebraic group defined over a finite field   $\mathbb{F}_q$   of   $q$   elements and   $\textbf{B}$   be a Borel subgroup of   ${\textbf{G}}$   defined over   $\mathbb{F}_q$  . Let   $\mathbb{k}$   be a field and we assume that   $\mathbb{k}=\bar{\mathbb{F}}_q $   when   $\textrm{char}\ \mathbb{k}=\textrm{char} \ \mathbb{F}_q$  . We show that the abstract-induced module   $\mathbb{M}(\theta )=\mathbb{k}{\textbf{G}}\otimes _{\mathbb{k}\textbf{B}}\theta $   (here   $\mathbb{k}\textbf{H}$   is the group algebra of   $\textbf{H}$   over the field   $\mathbb{k}$   and   $\theta $   is a character of   $\textbf{B}$   over   $\mathbb{k}$  ) has a composition series (of finite length) if   $\textrm{char}\ \mathbb{k}\ne \textrm{char} \ \mathbb{F}_q$  . In the case   $\mathbb{k}=\bar{\mathbb{F}}_q$   and   $\theta $   is a rational character, we give a necessary and sufficient condition for the existence of a composition series (of finite length) of   $\mathbb{M}(\theta )$  . We determine all the composition factors whenever a composition series exists. Thus we obtain a large class of abstract infinite-dimensional irreducible   $\mathbb{k}{\textbf{G}}$  -modules.

Abstract-Induced Modules for Reductive Algebraic Groups With Frobenius Maps