*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 the coset equality test 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. - Definition of a
*simple*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 . - Definition of
*conjugacy classes.*

### 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.

Can you prove Corollary 48 directly from the definitions, without using Proposition 47?

Pick some subgroups of groups. Which are normal in their respective groups?

### 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 2, 2016 at 10:12 am

[…] Expositions of interesting mathematical results « Groups and Group Actions: Lecture 10 […]

May 3, 2016 at 8:49 pm

Why are they called “normal”? Is there any connection to the geometric sense of the word and orthogonality?

May 4, 2016 at 8:03 am

Good question! Here’s a possible answer — does this help?

http://math.stackexchange.com/questions/898977/why-are-normal-subgroups-called-normal