Math 581    Problem Set 2    Due April 15

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

1   3   5   7

9 11 13 15

14 12 10 8

6   4   2  B

How about the following? 15 14 13 12

11 10   9   8

7   6    5   4

3    2   1   B

2. Let H be a nonempty subset of a group G.   Prove that H is a subgroup of G if and only if H itself with the multiplication "inherited" from G is a group. (By "inherited" is meant that if h, k  are in H, then the product hk in H is the element hk given by G's multiplication.)

3. Let G be a group and let g be any fixed element in G. Define a function  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 S6 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.