notes / Studies in Algebra and Group Theory / chapter 1: Sets and Groups / From my notebooks
N13 Things I forgor: cosets, normal subgroups, quotients
Left cosets
all the for . These lie in , but not in (when ): it’s , evil .
Yay! Right cosets are the SAME, but opposite.
Index of subgroups
Definition N13.1. The index of is the number of cosets (right or left, because both have the same number).
Normal subgroups
Definition N13.2. ( is normal) and for all , i.e., .
Kernel
For , . If :
where and is a pre-image/inverse image(?). Else, .
So we get the theorem:
Theorem N13.3. Given a group and , the following are equivalent:
Injections both ways
Theorem N13.4 (Schröder–Bernstein (finite case)). There are injections and .
Proof. () as sets (there is a bijection), and a bijection and its inverse give injections and .
() For finite : gives and gives , so . ∎
Scratch: .