*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.

### Further reading

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. There’s also a more recent hedgehogmaths video.

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.

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