# Permutation group order of operations and exponent

I ran across a problem in my homework that I am not quite sure how to approach. It is $((154)(13))^{13}$. Is this standard order of operations where I can evaluate within the parentheses before exponentiating? Also, is there an efficient way to evaluate this without say evaluating this step by step until you hit the 13th power? For the sake of this problem, we evaluate from right to left. Thanks!

• Can you work out the order of the permutation inside the brackets? – John Gowers Feb 24 '17 at 22:54

First, try to write $(154)(13)$ as a single permutation. That'll be $(1354)$.
Now, note that if you apply $(1354)$ four times, the numbers won't change places. That is, $(1354)^4=Id$. Can you continue from here?
$(154)(13)$ is not a disjoint cycle. their product will be a cycle of length 4.
$p^4 = e\\ p^{12} = e\\p^{13} = p$
Supposing they were disjoint, a $2$ cycle with a $3$ cycle would make a cycle of order $6$