In which we meet and explore quotient groups.
- Definition of the centre of a group.
- Proposition 51: Let be a group. Then . The proof is an exercise.
- Proposition 52: Let be a group, let be a subgroup of .
- Define a binary operation on via . This is well defined if and only if.
- If , then is a group.
Each part was a careful check, relying on the definition of a normal subgroup.
- Definition of a quotient group.
- Proposition 53: Let be a group, let be a subset of . Then is a normal subgroup of if and only if it is the kernel of a homomorphism with domain . One direction was Proposition 49. For the other, where so is a group, we defined the quotient map by , and showed that this is a homomorphism with kernel .
- Theorem 54 (First isomorphism theorem): Let , be groups, let be a homomorphism. Then , and the map given by is an isomorphism. We checked that is well defined (using Proposition 47), that it is a homomorphism, that it is injective, and that it is surjective.
- Corollary 55: Let be a finite group, let be a group, let be a homomorphism. Then . We used our proof of Lagrange’s theorem, which showed that if then .
Understanding today’s lecture
Can you prove Proposition 51 (that the centre of a group is a normal subgroup of that group)?
Can you fill in all the details in the proof of Proposition 52(ii) (that is a group)?
You could explore Corollary 55 by picking some explicit examples and seeing how it works for them.
Tim Gowers has written a really interesting blog post about normal subgroups and quotient groups. He’s also written about what it means to say that a function is well defined, which was very relevant for today’s lecture.
A long time ago, I wrote something about the first isomorphism theorem.
If, like me, you have recently found yourself trying to remember/discover how to write or in LaTeX, then you will appreciate Detexify.
Preparation for Lecture 12
How many homomorphisms are there from to ? (We’ll do this in the lecture, but do have a go before then.)
We’re also going to meet the notion of a group action. You could read this post by Tim Gowers to start to get a feel for what that’s about.