The class performance on the final examination ranged from a low of 40 to a high
of 146 on an examination with 150 maximum points (not including 15 bonus points available). The mean score of 96 (64%) was almost exactly the same as that for Exam 2, but * seven* students improved enough on the final to take advantage of the opportunity to improve midterm exam scores in the final grade calculation. For students whose percentage scores on Parts 1 and 2 of the final exceeded the mean of their two midterm examinations, the latter two grades were replaced by the *average* of the sum of the two midterm grades and the percent score on Parts 1 and 2 of the final exam.

Final course grades are available by clicking on the link Final Course Grades.

Late-breaking news about a study of the effects of cramming, as reported by the Hartford Courant Wednesday, May 15. Get a good night's sleep before the final exam!

Specially selected Review Problems for Final Exam preparation are now available.

** Final Exam** is scheduled for *Thursday, May 17, from 8:00 to 10:00 AM in MSB 307.*

Consultation hours prior to final exam, all in MSB 218*.

*In the event of a large turnout, the location will move to a classroom, and a note will be posted on the door of MSB 218 to specify which one.

The Exciting Summer Opportunities page has a new listing from People's Bank.

Two interactive Maple notebooks have been added to the course folder in MSB 203: * Groups and Cosets* and * Normal Subgroups*.

* Note*: The Course Information Sheet now shows the * correct * normal office hours.

** Hints: **

**Exercises 7 and 11(a), Section 8**: the calculations in class for *S*_{3
} enable you to show that *S*_{3} is not abelian and to determine its center. In doing the latter, it may be helpful to construct the Cayley multiplication table of *S*_{3}.

Next, note that every element of *S*_{3}, such as *f* = (1 2 3), is also in every *S _{n}* for

** Exercise 19, Section 5. ** Divide the problem into two pieces: (a) Show that *
x ^{r}* and

(b) Every subgroup *H* of an infinite cyclic group generated by *x* is generated by *x ^{r}* for some positive integer

** Exercise 21, Section 5. ** Let *G* be a cyclic group of order * n*. Find a
condition on the integers * r* and * s* that is equivalent to the subgroup generated
by * x ^{r}* being a subgroup of the subgroup generated by

* Comments.* First, it is easy to dispose of the case * r* = 0. So suppose without losing any generality that 0 < *r < n.*
The subgroup generated by * x ^{r}* will be contained in the subgroup generated by

* Last updated by J. Hurley 5/16/2001. *