I am reading a paper from Diaconis and Thiem about supercharacters and wanted to ask the following question:
In the proof of Lemma 4.1(a) I don't understand how to get from the third to the fifth line, i.e. why does the equality
$x_A(\phi_A)x_\beta(\phi(\beta))x_B(\phi_B)x_\beta(t)-x_\beta(t)=x_A(\phi_A)x_\beta(\phi(\beta)+t)[x_B(\phi_B),x_\beta(t)]-x_\beta(t)$
hold?
Thank you for your help.