Mathematical Scratchpad

1  Mathematics 433-A  Fall 2000

1.1   Exam 1

October 6, 2000

[`(                   Name                  )]

Technology used:                                                                                    

Directions:    

       Be sure to include in-line citations, including page numbers if appropriate, every time you use a text or notes or technology. Include a careful sketch of any graph obtained by technology in solving a problem.   Only write on one side of each page.

The Problems

1. (6 points each) Give the definitions of the following.
   1. The general linear group of order n over the real numbers GL( n,R) .(Include the binary operation.) 

   2. The center of a group G, Z( G) .

   3. The centralizer of an element a in a group G.

   4. A normal subgroup N of a group G.

   5. A homomorphism from the group G  to the group G ^¢.

2. (5 points each) Give examples of the following.
   1. A group that is not abelian.

   2. A finite, nontrivial, abelian group.

3. ( 15 points) Use one of the principles of mathematical induction to prove if a,b,c are elements in a group G for which b = cac^-1, then bⁿ = caⁿc^-1 is true for all positive integers n.

4. ( 15 points) Given a group G, subgroup H of G and element g Î G,we define the conjugate subgroup of H in G to be the set
   

   gHg-1 = { ghg-1:h Î H} .

   Prove gHg^-1 is indeed a subgroup of G.

5. ( 15 points) Given a homomorphism f:G® G ^¢ and a subgroup H of G, prove
   

   ker( f|H) = ker( f) ÇH.

6. ( 15 points) Do any one of the following.
   1. Prove that, in any group, the order of the product ab is the same as the order of the product ba.

   2. If G contains exactly one element of order 2,prove that element is in the center of G. [Hint: consider conjugates of that element.]

   3. Let G be an abelian group of odd order. Prove the map f:G® G defined by f( x) = x² is an automorphism.