92.421/521 Abstract Algebra

Fall 2011

Problem Set #5

Due November 21

Instructions:

Do the first two problems and any two of the last three problems.

Do each problem on a separate sheet of paper.  On each sheet, write your name And problem statement (it can be abbreviated). Include all logical steps/observations.

1.   Find all the subgroups of (Subscript[D, 4]). Which subgroups are normal? What are all the factor groups of (Subscript[D, 4]) up to isomorphism?

2.    Let  φ:ℤ→ℤ be given by φ(n)=7n. Prove that φ is a group homomorphism.  Find the kernel and the image of φ.

3.  Let G be the additive group of real numbers. Let the action of θ ∈ G on the real plane (ℝ^2) be given by rotating the plane counterclockwise about the origin through θ radians. Let P be a point on the plane other than the origin.
(a) Show that (ℝ^2) is a G-set.

(b) Describe geometrically the orbit containing P.

(c) Find the group (Subscript[G, P]).

4.  Let G = (Subscript[A, 4]) and suppose that G acts on itself by conjugation; that is, (g,h) → g h (g^(-1)).

(a) Determine the conjugacy classes (orbits) of each element of G.

(b) Determine all of the stabilizer (isotropy) subgroups for each element of (Subscript[G, x]).

5.  Find the number of ways a six-sided die can be constructed if each side is marked differently with 1, 2,…, 6 dots.