MAT 150A Study Guide - Final Guide: Surjective Function, Group Homomorphism, Bijection
1 May 2019
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G nite group, h g a subgroup. (lagrange) |g| Denote [g : h] called the index of h in g. To show lagrange we built an equivalence relation x y xy 1 h. We observed that [x] = hx. and each one has the same number of elements as h. G mod h = {set of left cosets} H\g := g(lef t) mod h (12) e (132) (13) (123) (23) H\g = {{e, (12)} , {(13), (132)} , {(12), (123)}} Let : g h be a surjective homomorphism. De ne : ker g h. If kx = ky = xy 1 k. Let h h, since is surjective, x g such that (x) = h. Let kx, ky k g such that (kx) = (ky) = (x) = (y) = xy 1 k, xy 1 k x = ky ky x = ex kx so x kx ky. = kx ky 6= = kx = ky.
