1. Prove``The left and right cosets partition G into equal sizedchunks." (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 thebook. Similarly so you have to show phi is 1-1 and onto, you can'tjust 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, weargue that all of the cosets have the same size)
(Corollary 5.13. Let G be a group and let H ? G. Then all of theleft 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 auseful characterization of cosets. Each part can either be proveddirectly or by appealing to previous results in this section.)
2. Use the above theorem to prove Lagrange's theorem. (Don't usea proof you read online or in the book, your goal is to prove itusing what you know about cosets).
