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Proposition: The signature of any r - cycle in S n is equal to ( - 1 ) r - 1 . Proof: Note that

Proposition: The signature of any r-cycle in
Sn is equal to (-1)r-1.
Proof:
Note that a typical r-cycle (a1a2cdotsar) 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={a1,dots,an} consider the
cycle f=(a1dotsar).
What properties does the cycle f have?
A.f has length r-1.
B.f is a permutation of S.
C.f sends a1a2,dots,ar-1ar
and ara1.
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,hinSn,sgn(gh)=sgn(g)sgn?
d. Where does the cycle on the LHS send a1?
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