MATH:5000 - Fall 2016

Homework Assignment 4

Reading (from Dummit and Foote):

Read Sections 3.4, 3.5, 4.1, 4.2.

Please look through the sections for EACH lecture before the lecture. See the Lecture Schedule for the sections that will be covered.

Then after EACH lecture, review your lecture notes and thoroughly read the sections covered in class.

SCHEDULE CHANGE in the week of September 19:

Monday (09/19): Lecture as usual.
Tuesday (09/20): Prof. Bleher teaches a lecture during the discussion session in 66 SH. Part A Problems are due as usual.
Wednesday (09/21): Erik Hasse teaches a discussion session during the lecture in 210 MLH.
Friday (09/23): Lecture as usual. Part B Problems are due as usual.

Part A Problems (to be handed in on Tuesday, September 20, at the beginning of the lecture):

Section 3.4: # 1, 6.
Hint for #1: Recall that Z_p is Dummit and Foote's notation for a cyclic group of order p that is written multiplicatively.
Section 3.5: # 3, 4.
Hint: Use that we proved in class that S_n is generated by all transpositions.
Section 4.1: # 1.

Part B Problems (to be handed in on Friday, September 23, at the beginning of the lecture):

If you hand in ALL Part B Problems together with the Part A Problems already on Tuesday, September 20, 2 extra points will be added to your homework score.

Section 3.4: # 9.
Hint for #9: First argue that r is at least 2. Then apply the second isomorphism theorem to the groups N_r-1 and M₁. Note that it may be useful to prove that if A, B are two normal subgroups of a group G, then AB is also normal subgroup of G.
Section 4.1: # 7(a)(b), 9, 10.
Hint for #9: For part (b), write |A| in two different ways, once using the transitive action of G on A and second using that O₁,...,O_r partition A.

Additional Practice Problems (you have to know how to do these, but you do NOT have to hand these in):

Section 3.4: # 4, 7.
Section 3.5: # 5, 6.
Section 4.1: # 2, 3, 4, 8.

Frauke Bleher
Sep 13 2016