Math 330 - Homework

All homework should have your full name (written legibly, not your signature) on all pages, and the pages must be stapled together, with the pages numbered, and the first page should say how many pages there are (write "page 1 of 7" for example).

The TA/grader for the course is Alexander Stathis.  He is posting solutions to the homework at:
http://homepages.math.uic.edu/~astathis/#teach-M330F2012
and he holds office hours are T/Th 10-12 in the Math Learning Center.

•     Week 1 (Due Wednesday September 5 -- Monday is Labor Day);  Review Chapter 1 and 2.1.  Do all problems except for Questions 2 a) and 5 from the sample Math 215 final from Fall 2000.
•     Week 2 (Due Monday September 10)  Read Sections 2.2 and 2.3.  Hand in:
• 2.19, 2.22, 2.27, 2.31
•     Week 3 (Due Monday September 17) Read Section 2.3.  Hand in:
  
• 2.36, 2,39, 2.40, 2.44
•     Week 4 (Due Monday September 24) Read Sections 2.3 and 2.4.  Hand in:
• 2.42, 2.46, 2.48, 2.49, 
•     Week 5 (Due Monday September 31) Read 2.3 and 2.4.  Hand in:
• 2.50, 2.43, 2.41, 2.51
•     Week 7 (Due Monday October 15)  Read 2.4 and2.5, Hand in:
• 2.52, 2.55, 2.58, 2.59
•     Week 8 (Due Monday October 22)  Read  2.5 and 2.6, Hand in:
• 2.67, 2.75, 2.76, 2.78, 2.92, 2.93
•     Week 9 (Due Monday October 29)  Read 2.6 and 2.7, Hand in:
  
• Recall that a group in which every element satisfies x² = e is Abelian.  Prove that this is not necessarily true if we replace 2 by any other prime p, as follows.  Consider the group with generators {a, b, c} in which a^p = b^p = c^p= e, a and b commute, a and c commute, and cb = abc.  Show that this is indeed a group, with order p³, in which every element has order p, and that it is not abelian if p>2.  Note:  it is enough to show that the set of all products aⁱb^jc^kwith 0 ≤ i,i,k < p, is closed under composition, and satisfies the associative rule.  The key point is that the composition law has to be consistent with c^p= e.
• 2.95, 2.103, 2.108, 2.113
  
•     Week 10 (Due Monday Nov 5)  Read 2.6 & 2.7, Hand in:
• 2.96, 2.98, 2.100, 2.109, 2.110
  
•    Week 13, Due Wednesday 11/28
  
• Prove that a group of order 27 cannot have an automorphism of order 5.
•    Week 14 Due Wednesday December 5
• 2.133 (i), 2.134, 3.2, 3.7, 3.19, 3.30