Question

1. Prove``The left and right cosets partition G into equal sized chunks." (Cor 5.11 and 5.13 in your book). You have to show the ~ is an equivalence relation, you can't just cite a theorem from the book. Similarly so you have to show phi is 1-1 and onto, you can't just cite a theorem from the book.

(Corollary 5.11. If G is a group and H ≤ G, then the left (respectively, right) cosets of H form a partition of G. Next, we argue that all of the cosets have the same size)

(Corollary 5.13. Let G be a group and let H ≤ G. Then all of the left and right cosets of H are the same size as H. In other words #(aH) = |H| = #(Ha) for all a ∈ G. † The next theorem provides a useful characterization of cosets. Each part can either be proved directly or by appealing to previous results in this section.)

2. Use the above theorem to prove Lagrange's theorem. (Don't use a proof you read online or in the book, your goal is to prove it using what you know about cosets).

Answer 1