MAT 313 Abstract Algebra  Fall 1999

Text:  J. Gallian Contemporary Abstract Algebra

Homework week VIII

• Problems  (due Tuesday Oct. 26 before the lecture)

 

 

Chapter	Exercises
9	4,10,12,26,44
10	6,16,26,36

Note: problem 10.30 has been cancelled!
 

• Reading Assignments

 

 

Chapter 11

Sections on Fundamental Thm.  and Isomorphism classes

Suggested further exercises

Solutions to homework week VIII

Exercise 9.26

G=DirectProduct[Z[4],Z[4]];
H=Closure[G,{{0,0},{2,0},{0,2},{2,2}}];
K=Closure[G,{{1,2}}];

Cayley Table for Z[4] Cayley Table for Z[2](+)Z[2]

Exercise 9.44

a) The Cayley table for the quaternion group G is printed below.
   Note that i is denoted by I, j by JJ and k by KK.

b) The left cosets of H in G are
   {1,-1} {I,-I} {J,-J} {K,-K}
   If you color, in the Cayley table for G,
   the elements of the first coset yellow,
   the ones of the second coset orange,
   the ones of the third coset purple and
   the ones of the fourth coset blue,
   then you notice that each coset appears in exactly ONE color.
   This is characteristic to normal subgroups as is explained
   in examples 9 and 10 in chapter 9.

c) The Cayley table for G/H, where H={1,-1}, is printed below.