STRANGE ALGEBRA

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Part 1 –(10 points each)

In order to receive full credit on each of the following problems you must show all work and provide explanation of your strategy where appropriate.

1. Let f: be given by f(x) =

a. Prove or disprove that f is onto,

b. Prove or disprove that f is one-to-one,

c. Prove or disprove that f(x1 + x2) = f(x1) f(x2),

d. Prove or disprove that f(x1 x2) = f(x1) f(x2).

2. Let S be the set of three elements given by S = {A, B, C} with the following table.

a. Is the binary operation * commutative? Why?

b. Determine whether there is an identity element in S for *.

c. If there is an identity element, which elements have inverses?

3. Find the greatest common divisor for a = 143, b = 385, and c = -65 and write it in the form ax + by + cz for integers x, y, and z.

4. Find a solution , for the congruence (mod 53).

5. Given [x] = [a]-1[b] is the unique solution in n to the equation [a] [x] = [b] solve the following by finding [a]-1 and [x] for [8] [x] = [7] in 11.

6. Complete the multiplication table for the group G = {a, b, c, d}

7. Compute gfg-1, the conjugate pair of f by g for f = (2, 3, 5, 4) and g = (1, 3, 2)(4, 5).

8. Find all normal subgroups of the octic group.

9. Consider the set R = {[0], [2], [4], [6], [8]} 10.

a. Construct addition and multiplication tables for R, using the operations for 10.

[a] +[b] = [a + b]

[a] ∙ [b] = [ab]

b. Observe that R is a commutative ring with unity [6], and compare this unity with the unity in 10.

c. Is R a subring of 10? If not, give a reason.

d. Does R have zero divisors?

e. Which elements of R have multiplicative inverses?

10. For f(x) = 2x3 + 7x + 4, g(x) = 4x2 + 4x + 6, and h(x) = 6x2 + 3 in 9 find f(x) + g(x)h(x) with all coefficients in 9.

Part 2 – (10 points each)

For each of the following, outline the strategy you would use to prove the given statement. For example, you might choose to use counterexamples, contradictions, deduction, and induction, direct or indirect proof and explain why you have chosen this strategy. In addition, please note the result you would need to be able to show.

IT IS NOT NECESSARY TO PROVE THESE

1. Let a, b, and c be integers. Prove or disprove that a|b implies ac|bc.

2. Let p be a prime integer. Prove that if [a][b] = [0] in p then either [a] = [0] or [b] = [0].

3. For an arbitrary set A, the power set (A) = {X | X A}, and addition in (A) defined by

X + Y = (X Y) - (X Y) = (X – Y) (Y – X)

a. Prove that (A) is a group with respect to this operation of addition.

b. If A has n distinct elements, state the order of (A).

4. Prove that a subset H of a finite group G is a subgroup of G if and only if

a. H is nonempty, and

b. a H and b H imply ab H.

5. Let G be a group of order pq, where p and q are distinct prime integers. If G has only one subgroup of order p and only one subgroup of order q prove that G is cyclic.

6. For each a in the group G, define a mapping ta: G→ G by ta(x) = axa-1. Prove that ta is an automorphism of G.

Part 3 – (20 points each)

Complete the proofs for any two of those outlined in Part 2. In order to receive full credit for these proofs you must include the proof and a reflection on the process of writing the proof. Your reflections should address any mathematical challenges you encountered and how these were addressed. (These are the “sticky” points in the proof. How did you handle those?)