Worksheet 11.3.1 Activity: Decomposing with Direct Products
Recall that last semester we saw that When does this sort of thing happen?
1.
Given positive integers and is it always true that If this is not always true, for which and is it true? Try some (many) examples.
2.
Consider Can we break this down as the direct product of two smaller groups? In other words is for some values of and
3.
Suppose your absent minded professor claims the answer is “no” and you don't feel like arguing. Maybe we can do something similar. Find two subgroups of call them and such that and In general, here it would be better to write
For any the group is the set of all positive integers less than and relatively prime to under multiplication modulo For example we saw that is a group under multiplication modulo 8.
Consider the group The table below gives the twelve elements with their orders:
|
1 |
3 |
5 |
9 |
11 |
13 |
15 |
17 |
19 |
23 |
25 |
27 |
|
1 |
6 |
6 |
3 |
6 |
2 |
2 |
6 |
6 |
6 |
3 |
2 |
4.
Let be the set of all elements of order for some (that is, elements with order some power of ). Find and for
5.
Are and subgroups of
6.
Do and have the property that and
7.
Is Is for some values of and