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?
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?