March 28

I've been asked for some study aids over break. Here's what I've come up with.

First and foremost, you should know all the definitions that we've made in class and used more than once. There are only about twenty of these, I think. One strategy is to recopy them onto a Definitions Page that you then rarely look at again.

Go through the theorems we've proved and look at the assumptions we made. For each such assumption, knock it out and find a counterexample to the theorem. (In less well-developed areas of mathematics, it's often the case that people have assumptions that everyone believes are unnecessary, but noone knows a counterexample without the assumption.)

One of the great benefits of studying finite groups, as opposed to e.g. uncountable subsets of the real line, is that it's easy to pound through examples in absolutely complete detail. If some theorem looks weird to you then try it out, one by one, on the small groups you know (e.g. the groups of order at most 10, S_4, A_5).

Here are some easy book questions that you should regard as handwriting exercises. If these aren't easy, then you definitely have something to study.

0.2: #3,5

0.3: #3,9

1.1: #5,8,9,10,11,12,16,20,21,23,29

1.2: #1,4

1.3: #1,2,3,4,6,7,8,13,14

1.6: #1,2,8,9,13,15,16,17,18

1.7: #3,11,12,14

2.1: #2,3,5,14,15

2.2: #2,7,8

2.3: #1,2,3,11,12,19,25 -- that one is just a little harder

2.4: #1,2,3,4,5,6,8,13,16

Now we're getting into richer stuff, so these are still easy, but not all at the level of handwriting exercises

3.1: #1,2,3,4,5,6,7,8,9,10,12,16,17,21,22,24,27,36,39

3.2: #1,5,6,7,8,12,14,17,18 (harder)

3.5: #4 (harder)

4.1: #4 (we did),5,8a

4.3: #2,3,7,18,27,30 (harder but cute)

4.4: #15,18 (much harder but cool)

4.5: #1,2,3,4,5,6,7,8,13,17,18,30 (easy only once you've done homework #8)