In which we think about the link between equivalence relations and partitions, and meet cosets.
- Definition of an equivalence class.
- Definition of a partition of a set.
- Theorem 27: Let be an equivalence relation on a set . The equivalence classes of partition . To prove this, we checked that each equivalence class is non-empty, that between them they cover the set (their union is the set), and they are pairwise disjoint, using each of reflexivity, symmetry and transitivity along the way.
- Theorem 28: Let be a partition of a set . For , write for the unique part in with . Define a relation on by if and only if . Then is an equivalence relation. We checked reflexivity, symmetry and transitivity.
- Corollary 29: There is a bijection between equivalence relations on a set and partitions of the same set . This was a consequence of Theorems 27 and 28.
- Definition of , and of binary operations and on .
- Lemma 30: The operations and on are well defined. Once we’d worked out what we needed to worry about, it was straightforward to check that in fact we didn’t need to worry.
- Proposition 31: is an Abelian group. Moreover, it is cyclic and isomorphic to . Furthermore, is associative and commutative on , and is distributive over . The proof of this is an exercise.
- Proposition 32:
- Take . Then has a multiplicative inverse in if and only if .
- If is prime, then is a field.
- Let has a multiplicative inverse be the set of units in . Then is a group under multiplication. The proof is an exercise on Sheet 4.
- Definition of a left coset of a subgroup in a group . Definition of the notation for the set of cosets. Definition of the index of a subgroup in a group. Definition of right cosets.
Understanding today’s lecture
Can you prove Propositions 31 and 32?
What are the left cosets of in ? What is the index of in ?
Pick a group and a subgroup. What are the left cosets? For example, you could pick a small dihedral group like (the symmetries of a square) and explore left cosets of subgroups of .
In Lemma 30, we showed that two functions we’d just written down were well defined. What does well defined mean? Here are some useful thoughts from Tim Gowers.
I’ve written before about the result that the non-zero integers modulo a prime form a group under multiplication, which is closely related to (for prime) being a field.
If you enjoy reading blog posts relating to your studies, then you should look at Eventually Almost Everywhere, a blog by Dominic Yeo, who’s been tutoring the Linear Algebra II course recently.
And here’s something a bit lighter, in case you’re feeling old as the end of term approaches.
Preparation for Lecture 8
Let and be isomorphic groups, and let be an isomorphism. Take . What can you say about the orders of in and in ? How are they related?
Can you say anything interesting about the index of a subgroup in a finite group , in terms of the orders of and ?
Take a finite group , and a subgroup . Can you show that the left cosets of partition ? What is the corresponding equivalence relation on the set ?
Can you show that all the left cosets of in have the same size?