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?
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.
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?