*In which we think some more about homomorphisms, and meet normal subgroups.*

- Proposition 45:
*Let be a homomorphism. Then there is with for all .*We proved this by defining and then using the fact that generates . - Definition of the
*kernel*and*image*of a homomorphism. - Proposition 46:
*Let , be groups, let be a homomorphism. Then**is a subgroup of ; and**is a subgroup of .*

We proved this using the subgroup test.

- Proposition 47:
*Let , be groups, let be a homomorphism. Then is constant on each coset of , and takes different values on different cosets.*We saw that if and only if , and then used Lemma 33 to see that this is equivalent to . - Corollary 48:
*Let , be groups, let be a homomorphism. Then is injective if and only if .*This was immediate from Proposition 47, since . - Definition of a
*normal subgroup*of a group. - Proposition 49:
*Let , be groups, let be a homomorphism. Then is a normal subgroup of .*We already know from Proposition 46 that is a subgroup of , so we just checked that if and then . - Proposition 50:
*Let be a subgroup of a group with index . Then .*We argued that the only (left and right) cosets of in are and , and used this to see that for all .

### Understanding today’s lecture

Pick some homomorphisms. Can you identify their kernels and images? Which homomorphisms are injective? Which are surjective?

Pick a homomorphism between two groups (pick explicit groups and an explicit homomorphism). What are the cosets of ? This might help you to get a feel for Proposition 47.

Pick some subgroups of groups. Which are normal?

### Further reading

Of course Wikipedia has a page about normal subgroups, and another about the notion of a simple group. You will learn more about simple groups if you choose to study Algebra 3 (Group Theory) next year.

MacTutor suggests that Galois was the first to recognise the significance of normal subgroups.

### Preparation for Lecture 11

Can you show that if is a normal subgroup of then (the set of left cosets of in ) forms a group under the natural operation?

May 4, 2015 at 10:27 am

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