How do you prove (by means of induction) that the following is true for all group-like algebraic structures?
$$\operatorname{ord}(a_1 \circ a_2 \circ a_3 \circ \cdots \circ a_{n-1} \circ a_n) = \operatorname{ord}(a_2 \circ a_3 \circ \cdots \circ a_{n-1} \circ a_n \circ a_1)$$
where ord refers to the order of the group.
$(a_1a_2 \dotsb a_n)^{m+1} = a_1(a_2 \dotsb a_na_1)^ma_2 \dotsb a_{n-1}$ is all you need.