301 Abstract Algebra
2:00 - 2:50 MWF
Hentzel 432 Carver
hentzel@iastate.edu
Web Page   http://www.math.iastate.edu/hentzel/class.301

Text: A First Course in Abstract Algebra, sixth Edition 
by John B. Fraleigh

Course: Chapter 1,2,3,4,5 
  
Wednesday, October 17, 2001 

**********************************
*   TEST FRIDAY October 19       * 
*    pages 161 through 177       *
*    pages 209 through 210       *
*    pages 217 through 229       *
**********************************

Main Idea:  Study these problems for a good review
            for the test. 
Key Words:  What-ever-else you know, know this. 
Objective:  Focus your attention from the myriad of
            stuff we covered to the critical mass for
            Friday. 


1.  How would you weigh 17 cups using 

a 19 cup measure and a 29 cup measure?



2.  (a)  Show that if GCD(x,n) = 1, then x has

an inverse mod n.


    (b) Find the inverse of  321 mod 89.


3. In the RSA code, n = 13*17 and r = 5.  What is the

decoding exponent.



4.  (a) State the first Sylow Theorem.   

    (b) Prove the first Sylow Theorem.

    (c) State the second and third Sylow Theorem.

    (d) Prove the second and third Sylow Theorems.



5.  Give the possible numbers of p-sylow subgroups

for a group of size 5*7*9.










6.  For the following puzzle peg board,  if it

could be reduced to one peg by horizontal and

vertical jumps, where must the last peg be

located?     (o) = peg,  ( ) = hole

        (o) (o) (o) (o)
        (o) (o) (o) (o)
        (o) (o) (o) (o)
        (o)         ( ) 


7.  Find a permutation pi such that

                 -1  
pi (123456789) pi   =   (987654321)




8.  Suppose that G is a group and H is a normal

subgroup of G.

(a)  What is the definition of a normal subgroup.

(b)  Prove that Hx*Hy c H(xy) for all x,y in G.


9.  Suppose that f:G-->H is a homomorphism of group

G into group H with kernel K..  

(a)  What is the definition of Ker(f).

               
(b)  Show that f(a) = f(b) if and only if Ka = Kb


10. Suppose that f:G-->H is a homomorphism of

group G into Group H with kernel K.  We define a map  

g(Ka) = f(a).   

(a)  Explain why one might suspect that g is not

really defining a function.

(b)  Prove that the map g is well defined.




 



 

(b)