1. Explain why the following position cannot be reached from the original configuration of the 15-puzzle.

9 11 13 15

14 12 10 8

6 4 2 B

11 10 9 8

7 6 5 4

3 2 1 B

3. Let G be a group and let g be any fixed element in G. Define a function f from G into G by

"h in G, f(h) = g'hg.

Prove that f is an isomorphism
of G onto G. (An isomorphism of a group G onto itself is usually called
an **automorphism** of G. This particular automorphism is usually called
conjugation by g.)

4. Let S_{6 }denote
the group of all permutations of the set {1, 2, 3, 4, 5, 6}.

Let g = (1 4 5 3). Calculate g'hg for the following permutations
h.

(a) h = (2 6 4)

(b) h = (2 6 5)

(c) h = (2 6 1)

(d) h = (2 4 5)

(e) h = (2 5 4)

(f) h = (3 4 1 2 6 5)

(g) h = (6 2)(4 5)

(h) Formulate a general rule which encompasses
these calculations. State it carefully, as if you were writing it for inclusion
in a textbook.

5. Textbook p. 206, #3

6. Textbook, p.206 #1 modified. Instead of showing
that this system forms a semigroup, show that it does *not* form a
group.