Proposition: The signature of any $r-$cycle in

$S_n$ is equal to $(-1{)}^{r-1}.$

Proof:

Note that a typical r$-$cycle $(a_1{a}_{2}cdots{a}_r)$ ci

We know that the signature of the r$-$cycle on

Note that there are

$2-$cycles on th

$5.$ Therefore the signature of the $r-$cycle must $b$

a$.$ Fill in the gaps in the proof.

b$.$ For a set $S=\{a_1,$dots,$a_n\}$ consider the

cycle $f=(a_{1}dots{a}_r).$

What properties does the cycle $f$ have?

A$.f$ has length $r-1.$

B$.f$ is a permutation of $S.$

C$.f$ sends $a_1{a}_2,$dots,$a_{r-1}{a}_r$

and $a_r{a}_1.$

D$.f$ is an onto $($i$.$e$.$ surjective$)$ function on

$S.$

E$.f$ has a signature of $-1$

c$.$ Which step of the proof used the fact that for

any

$g,hin{S}_n,sgn(gh)=sgn(g)sgn?$

d$.$ Where does the cycle on the LHS send $a_1?$