Spring 2006

100 points possible.

1. (20 pts.) Consider t = (1 4 2 8)(2 8 6)(8 6 5 3) Î S₈.

(a) Write t in permuation notation.

(b) Write t as a product of disjoint cycles.

(c) Write t as a product of transpositions.

(d) Is t an even or odd permutation?

2. (20 pts.) The set G={1,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O} and the binary operation * given in the table on the supplemental page form a group (you do not have to prove this).

(a) What is the identity element in G?

(b) Find F^-1.

(c) Find áKñ.

(d) Find |áAñ|.

3. (20 pts.) Define a binary operation * on Z by letting a*b=ab-1 for all a,b Î Z.

(a) Is * commutative? Prove it, or give a numerical counterexample.

(b) Is * associatative? Prove it, or give a numerical counterexample.

4. (20 pts.) Let n ³ 2 and define H={ s Î S_n | (1 2)s = s(1 2)}. Prove that H £ S_n.

5. (20 pts.) Let n ³ 3 and define f: S_n® S_n by

f(s)=(1 2 3)s,   for all s Î Sn.

Prove that f is one-to-one and onto. (Recall that y:X® Y is one-to-one iff "x₁,x₂ Î X, if y(x₁)=y(x₂), then x₁=x₂. Recall that y:X® Y is onto iff "y Î Y, $x Î X such that y(x)=y.)