MAT 150A Study Guide - Final Guide: Coset, Normal Subgroup, Surjective Function

• School: UC-Davis
• Department: Mathematics
• Course: MAT 150A
• Professor: José Simental Rodríguez

Document Summary

Let h g be a subgroup, then h is normal a g, ah = ha. H =< (12) >= {e, (12)} eh = h (12)h = {(12), e} (13)h = {(13), (123)} (23)h = {(23), (132)} (123)h = {(123), (13)} (132)h = {(132), (23)} If h g is a normal subgroup denote g\h = {all cosets of h} Let s1, s2 g be subsets of g. de ne their product. S1 s2 = {g1g2|g1 s1, g2 s2} = {(13)(123), (13)(132), (123)(23), (123)(132)} = {(132), (23), (12), e} October 22, 2018 bc not one of the cosets as previously established. Then cosets ah, bh, the product (ah)(bh) is another coset, and moreover (ah)(bh) = (ab)h. Let (ah1)(bh2) (ah)(bh) abh = (ae)(bh) (ah)(bh) (ah1)(bh2) = a(h1b)h2. Since h is normal, hb = bh = h1b = bh a(h1b)h2 = ab(h h2) = abh (ah)(bh) = a(hb)h = a(bh)h = abhh = abh (cid:4) Let h g be a normal subgroup.