On Shalev's conjecture for type 𝐴𝑛 and 2𝐴𝑛

Abstract

In the paper, we consider images of finite simple projective special linear and unitary groups under power words. In particular, we show that, if G ≃ PSL ε n ( q ) , then, for every power word of type x M , there exist constants c and N such that | ω ( G ) | > c ln ( n ) | G | n whenever | G | > N .