Section 3.1 Cosets
LetExample 3.1.
Let
We will always write the cosets of subgroups of
Example 3.2.
Let
The right cosets of
It is not always the case that a left coset is the same as a right coset. Let
however, the right cosets of
Lemma 3.3.
Let
Theorem 3.4.
Let
Proof.
Let
Remark 3.5.
There is nothing special in this theorem about left cosets. Right cosets also partition
Example 3.6.
Let
Example 3.7.
Suppose that
Theorem 3.8.
Let
Proof.
Let
Again by Lemma 3.3,
Reading Questions Reading Questions
1.
Why do we sometimes need to specify left or right cosets instead of just calling them cosets? When do we not need to worry about this?
2.
Let
3.
Let
4.
After reading the section, what questions do you still have? Write at least one well formulated question (even if you think you understand everything).
Exercises Practice Problems
5.
List the left and right cosets of the subgroups in each of the following.
in in in in in in in in
(a) \(\langle 8 \rangle\text{,}\) \(1 + \langle 8 \rangle\text{,}\) \(2 + \langle 8 \rangle\text{,}\) \(3 + \langle 8 \rangle\text{,}\) \(4 + \langle 8 \rangle\text{,}\) \(5 + \langle 8 \rangle\text{,}\) \(6 + \langle 8 \rangle\text{,}\) and \(7 + \langle 8 \rangle\text{;}\)
(c) \(3 {\mathbb Z}\text{,}\) \(1 + 3 {\mathbb Z}\text{,}\) and \(2 + 3 {\mathbb Z}\text{.}\)
12.
If
Let \(g_1 \in gH\text{.}\) Show that \(g_1 \in Hg\) and thus \(gH \subset Hg\text{.}\)
17.
Suppose that
18.
If
Exercises Collected Homework
1.
Let
Remember you need to prove two directions: first assume \(aH = bH\) and prove that \(b\inv a \in H\text{.}\) Second, assume \(b\inv a \in H\text{,}\) and prove \(aH = bH\text{.}\) To prove this second direction, it would be enough to show that \(a \in bH\) using the fact we proved in class.