# Exam Feb 22, 2017

The exam will cover up to and including Section 4.2.

The bonus problems (not to be presented during the exam) are as follows:

• Given the category of finite abelian groups, can you determine which object is which group (up to isomorphism)?
• Determine the group generated by two adjacent faces of the Rubik’s cube.

# HW5, due Wed Feb 15, 2017

• DF 3.2 #8, #11,
• DF 3.3 #4, #7,
• DF 3.4 #5, #7, #11, #12

# HW 4, due Wed Feb 8, 2017

• DF Section 2.4 #9 #17 (Zorn’s lemma appears in Appendix 1), #18
• DF Section 3.1 #1, #12.
• Let F be any field.  Show that there is a 1-1 homormophism from $S_n$ to $GL(n,F)$ given by the action of $S_n$ on the indices i in the standard basis $e_1,\ldots,e_n$.  Combine this with Cayley’s theorem (page 120) to show that for every finite group G, there exists n, such that G is isomorphic to a subgroup of $GL(n,F)$.
• Show algebraically that every isometry T of $R^2$ is a glide reflection provided it has the form $\latex T=T_\lambda A$, where $A\in O(2,R)$, $A\not\in SO(2,R)$, $\latex \lambda\ne 0$.
• Solve the following problems without submitting a solution
• DF Section 2.4 #15, #19

# Midterm Exam

The midterm exam will be held on Wednesday, February 22, 2017.

# Office hours

Bill Rau it the TA for the course.  He is available to answer questions about homework in the MAC 10-12 Mondays and 4-5 Mondays.

# HW3, due Wed Feb 1, 2017

In class on January 25, we went over a large number of exercises.  You will not be asked to submit solutions on February 1.  Instead, read all of the exercises from Sections 2.1 (17 problems), 2.2 (14 problems), and 2.3 (26 problems).

Submit a sheet of paper stating for each problem which category each problem falls into:

• (Solved) You have solved the problem.
• (Understand)  You have not solved the problem, but you are convinced that you understand exactly how to solve the problem.
• (Unsolved) You are not convinced you understand how to solve the problem.

# HW2, due Wed Jan 25, 2017

• DF Sec 1.3 #5, #8, #14
• DF Sec 1.6 #9, #17, #23,
• DF Sec 1.7 #17
• (wording updated on Jan 23.) Let F be a field.  Let $V=F^n$, viewed as an n-dimensional vector space over the field F.  Show the following:
• The group G=GL(n,F) acts on V by linear transformations.
• If $B=(e_1,\ldots,e_n)$ is  the standard ordered basis of V and if $g \in G$, then $g B = (g e_1,\ldots,g e_n)$  is an ordered basis of V.
• There is a 1-1 correspondence between elements of GL(n,F) and bases of V.

# HW1, due Wednesday Jan 18

• DF Section 1.1, problems #5, #9, #19 (do this one carefully using induction), #33a
• DF Section 1.2, problems #2, #9, #16.

# Class starts Monday Jan 9.

The first day of class will be Monday January 9, 2017.

There will be no meeting on Wednesday Jan 4 and no meeting on Friday Jan 6.