for cyclic permutation $g=(i_1i_2\cdots i_p)$, prove $\operatorname{sign}(g)=(-1)^{(p-1)}$

Question

I'm reading "A course in algebra" by E. B. Vinberg for a basic understanding.

Now I met a problem in Exercise 4.99: Deduce the following formula for the sign of a cyclic permutation: $$\operatorname{sign}(i_1i_2\cdots i_p) = (-1)^{p-1}$$

I'm a bit lost here... how to prove it? Actually at the first step I'm not sure how to define a cyclic permutation. should it be defined that ${1,2,\cdots,p}$ forms a cyclic group, generated by $k$, such that $i_j = j\circ k$?

Answer 1

Hint:

$$(i_1\,i_2\,\ldots\,i_p)=(i_1\;i_2)(i_2\;i_3)\cdot\ldots\cdot(i_{p-1}\,i_p)$$

(Here, operation is function's composition: from right to left)