HOMEWORK

Simmons, G. F.
Mathematical rigor is like clothing; in its style it ought to suit the occasion, and it diminishes comfort and restrains freedom of movement if it is either too loose or too tight.
In The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.

#7:     6.1: 4, 6, 8(a,b,c, d), 12, 14
          6.2: 4, 7
          6.3: 3, 13(a)
          6.4: 1, 2, 15, 16
          6.5: 1, 2, 4           
Collected Friday 8/13: 6.1.4, 6.1.8(a&c), 6.1.12, 6.4.1, 6.4.15, 6.5.2

#1:     2.1: 1, 4, 7, 8, 11
          2.2:  3, 4, 7, 8, 10, 12-14, 16, 17, 20(a), 21
          Extra: Give an example of a non-associative operation on the real numbers. 
Collected Friday 6/25: 2.1.11, 2.2.4, 2.2.16(a), 2.2.17 (the "Hello world" problem of Group Theory)
 

#2:     2.3: 1, 4, 6(a,c), 7, 12, 14
          2.4: 3, 5 (what is the kernel?), 8, 10, 13, 16
          2.5:  5, 6, 7, 8(a,c)
          2.9:  1, 2(a), 3, 4
Collected Friday 7/2: 2.3.12, 2.4.8, 2.5.6, 2.9.4

#3:     2.6: 1, 3, 7, 9(a), 10
          2.10: 1(a), 4, 5, 8
          Extra: Classify all groups of orders less than or equal to 5, up to isomorphism. (List and describe them. Prove you got all)
Collected Wednesday 7/7: 2.6.3, 2.6.10, 2.10.4(a)(c), Extra problem.
Hints:
    2.6.3: Pick any element of G other than the identity. What can you say about its order? Use problem 2.2.10.
    2.6.10 (a): [G;H]=2 => G is a disjoint union of H and another coset, xH, for some x in G (but not in H).
                            Show xHx^-1ĚH. Show gHg^-1ĚH for all g in G.
               (b): an assigned problem from section 2.10 helps here. Which one?
    Extra: When the order is prime, a result in section 6 determines all possibilities. For the groups of order 4  you need more work, but (6.12) should help.

#4:       2.10: 1(a), 5, 7, 8
            2.7: 1, 2, 7
            2.8: 1, 2, 3, 5, 8, 9, 10         
Collected Friday 7/16: 2.10.1(a),  2.7.7, 2.8.2, 2.8.9.

#5:       5.1: 3 (a, b,d)
            5.3:  1, 4, 5
            5.5:  1(a&c), 3, 4, 5, 8 (note in #8: e₁ is the vector [1,0,0,...,0])
Collected Wednesday 7/28:  5.1.3 (a, b), 5.3.4, 5.5.1(a,c), 5.5.5

#6:        5.6: 1, 3(a)
             5.7: 1, 2, 3
             5.8: 3, 4
             5.9: Read and understand the content of Theorem (9.1). Compare and contrast with Theorem (3.4).
             6.1: 1,2
    Extra Problem #1:
             Classify all groups G of order 6 up to isomorphism.
                    a) What orders can the elements have?
                    b) Prove that such a group G must have at least an element x of order 3, by showing that it cannot have all but the
                        identity be of order 2.
                    c) Prove that such a group G must have at least an element y of order 2, by showing that it cannot have all but the
                        identity be of order 3.
                    d) Show that G is generated by x and y.
                    e) Show that there are only two possibilities for xy: either xy=yx (what group do you get?) or xy=yx²(what group do
                        you get then?) 

    Extra Problem #2:
             Consider the triangular prism in the picture (the ends are                
            congruent equilateral triangles).

a) What are all the rotation symmetries of this prism? What group structure do they have?          

b) What are all the symmetries that stabilize the end triangles? List them as elements of S₃ by considering their action on the triangle 123. What group do they form?

c) What are all possible symmetries? List them and identify the group they form in terms of familiar groups.

Collected Wednesday 8/4: 5.6.1, 5.7.2 and the Extra problems #1 and #2.