MAT 534 Problem Sets Algebra I Fall 2009

There will be weekly problem sets collected in the Thursday lecture. Problems due one week will often practice material covered the previous week. It is important to do the problems in a timely manner; do not postpone problems until the last minute.

You are encouraged to work with other students in the class, but your final write-up must be your own and must be based on your own understanding. Moreover, write-ups must be legible, etc. Write-ups which prove too difficult for the grader to read may be marked incorrect or may be returned to the student to be rewritten. All questions regarding grading of a problem set should be addressed to the instructor in a timely fashion.

In computing final grades, only the 10 best problem sets (out of 11 total graded problem sets) will be used.

In the assignments below, please work through all problems. Only the RED problems will be graded. Please direct any other questions regarding problem sets to your instructor.

• Problem Set 1 is due in the Thursday lecture the week of Sept. 7th — 11th.
  Section 1.1: 9, 22.
  Section 1.2: 4, 10.
  Section 1.3: 2, 13.
  Section 1.4: 2, 11.
  Section 1.6: 9 (Hint: Use Exercise 1.2.4 and 1.3.13 assigned above.), 23.
  Section 1.7: 8, 10.
  
• Problem Set 2 is due in the Thursday lecture the week of Sept. 14th — 18th.
  Section 2.1: 6, 10.
  Section 2.2: 4, 14.
  Section 2.3: 21, 26.
  Section 2.4: 7, 12.
  Section 2.5: 11, 16.
  
• Problem Set 3 is due in the Thursday lecture the week of Sept. 21st — 25th.
  Section 3.1: 17, 24.
  Section 3.2: 4, 12.
  Section 3.3: 3, 8.
  Section 3.4: 7, 8.
  Section 3.5: 3, 16.
  
• Problem Set 4 is due in the Thursday lecture the week of Sept. 28th — Oct. 2nd.
  Section 3.4: 9, 10.
  Section 3.5: 6, 8.
  Section 4.1: 3, 7(a) and (d).
  Section 4.2: 11, 12, 13.
  
• Problem Set 5 is due in the Thursday lecture the week of Oct. 12th — 16th.
  Section 4.3: 19, 21, 24.
  Section 4.4: 2, 10.
  Section 4.5: 24, 38.
  Section 4.6: 3, 4.
  
• Problem Set 6 is due in the Thursday lecture the week of Oct. 19th — 23rd.
  Section 4.6: 6, 7.
  Section 5.1: 7, 8*, 9*.
  Section 5.2: 8, 14.
  Section 5.4: 8, 9.
  * There is a small mistake in Problems 8,9 of Section 5.1. Please prove injectivity of the homomorphism from Problem 8 only for the special case occurring in Problem 9.
  
• Problem Set 7 is due in the Thursday lecture the week of Oct. 26th — 30th.
  Section 5.5: 7, 13, 23.
  Section 11.1: 6, 8, 9.
  Section 11.2: 19, 26, 33.
  
• Problem Set 8 is due in the Thursday lecture the week of Nov. 2nd — 6th.
  Section 11.3: 2, 3, 4.
  Section 11.4: 4, 5.
  Section 11.5: 5, 11, 13.
  
• Problem Set 9 is due in the Thursday lecture the week of Nov. 9th — 13th.
  Problem Set on the spectral theorem ( pdf, dvi, ps).
• Problem Set 10 is due in the Thursday lecture the week of Nov. 30th — Dec. 4th.
  Main Problem. Let G and G' be groups with the same finite order n. Let m be some integer divisor of n, and assume that G and G' each contain a unique normal subgroup P, resp. P', of order m. For each group, consider the associated group which is the centralizer of P, resp. P', modulo the center of P, resp. the center of P'. If G and G' are isomorphic, prove that these associated groups are also isomorphic.

  Next, assume that there exists a subgroup Q of G which intersects P in only the identity element and such that P and Q generate G, i.e., G is a semidirect product of P and Q. Let f denote the induced homomorphism from Q to the outer automorphism group of P (not the usual homomorphism to the automorphism group of P). Show that the kernel of f is canonically isomorphic to the quotient of the centralizer of P by the center of P. Thus the isomorphism class of the kernel of f is an isomorphism invariant of G. In particular, observe that the hypotheses hold if we restrict to groups G and G' which are each a semidirect product of a fixed finite p-group P and a fixed group Q whose order is less than p+1. Moreover, if P is abelian then the outer automorphism group of P equals the usual automorphism group of P so that f is the usual homomorphism.

  Second Problem. For every problem on Midterm 2 where you lost points, please write up a complete, correct solution to that problem (if you only lost points on a part of the problem, you can write up just the solution for that part). If you got full credit on the exam, you do not need to write up anything (you will automatically get credit for this part of the homework assignment).

• Problem Set 11 is due in the Thursday lecture the week of Dec. 7th — Dec. 11th.
  This problem set is to be assigned. Section 12.1: 16, 17, 18, 19 (Please read through this sequence of exercises.)
  Section 12.3: 2, 9, 13, 14, 17.
  

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