Exercise 1, § 1

Assume that $k$ has characteristic $p >0$. Let V be a vector space, S a finite set. A map $r: S\rightarrow$ End(V) satisfies condition (AC) if and only if there exists a power $q$ of $p$ such that $[s^q{, s'}^q] = 0$ for all $s, s'\in S$. (Use Chap. I, §1, Exerc. 19, formula (1).)