Lagrange Theorem states that the order (number of elements) of every subgroup of G is divisible by the order of G itself. It's easy to understand, yet compelling.
The issue is that I really don't know much about Group Theory, but I've been able to help myself somehow with the Internet. However, I've become stuck with a very simple Wikipedia proof of Lagrange Theorem. It has some vocabulary that I had to look up and learn, so maybe the issue is that I didn't comprehend some words...I don't know.
I'm thinking of quoting each sentence of the proof and explaining my understanding of it. I figure that in that way someone might find where I got something wrong. I would really appreciate any help/thoughts! If there is something that isn't clear from my explanation I would be totally happy to clarify.
"Proof of Lagrange's theorem
This can be shown using the concept of left cosets of H in G."
- H is a subgroup of G. A left coset of H is defined as gH, where g is an element of G.
"The left cosets are the equivalence classes of a certain equivalence relation on G and therefore form a partition of G."
- Since it says 'therefore', I believe that the significant part in the latter sentence is that the left cosets of H are a partition of G. Which would be true since for each g and H we choose, the elements of gH (because of the definition of a group) would be in G.
"Specifically, x and y in G are related if and only if there exists h in H such that x = yh."
- If this statement is more than a definition then I don't get it. I would understand that it is defining what conditions must a pair of elements have in order to be related.
"If we can show that all cosets of H have the same number of elements, then each coset of H has precisely |H| elements."
- If a coset has an element twice, then it would not have |H| elements, right? Or are they claiming that there exists no such thing as a coset with repeated elements?
"We are then done since the order of H times the number of cosets is equal to the number of elements in G, thereby proving that the order of H divides the order of G."
- Again, I'm confused. Isn't a coset defined as gH (g being an element of G)? So, since there are |G| elements we can choose from, there should be |G| left cosets of H...right? And for |G| times |H| to give |G|, |G |H| would have to be 1. This sentence also made me become sure that I likely got some definitions wrong.
"Now, if aH and bH are two left cosets of H, we can define a map f : aH → bH by setting f(x) = ba−1x. This map is bijective because its inverse is given by f−1(y) = ab−1x"
- And now they seem to assert that each coset of H has the same elements. Which -if it's true- just passes my mind.
Is it that I've got the definitions wrong? That would explain why I don't understand what they are saying. Or is it that I do understand what they are saying inly that their statements are simply clear for them but not for me?
I would truly, truly appreaciate any help.