Here are some things you may assume. there are proofs in the book but you don't have to write them down. we write h g to say that h is a normal subgroup of g and hg either of these conditions is equivalent to h g (along with h? g) .ah ha for all a eg 2. aha-1-{aha-1 lhe h} = h for all a g .if g is finite and p is a prime divisor of |g then g has at least one element of order p there is a table for as and some other small groups in the back of the book. you can use that we will extend the notation of the book a bit and say that for a ring r with unity 1r given an element ue r we say that v risa left inverse of ts if vu = 1r and that w e r is a right inverse if uw = 1 r the difference here is that r might not be commutative. we call u a unig if it has at least one left inverse and at least one right erse . similarly? #1) is a left zero divisor if there is some y? 0 e r with zy = 0 and a right zero divisor if there is either some y oe r with ya 0 we say that u is a zero divior iit is either a left zero divisor or a right zero divisor or both 1. let r be a non-commutative ri with unity and u be an element (a) prove that if u is a unit as described above, so there are v, w with (b) suppose that u has a right inverse with uzr show that if there is no other yr with uy 1r then u is not a zero divisor (c) suppose that u has a right inverse z with - rshow that if there is no other u? r with try = 1r then u is a unit. 2. consider the subset h = fe, ( 12)(34), ( 13) (24), ( 14)(23)) of g-a4 (a) show that h is a subgroup of g (b) write out all the left cosets of h in g (c) prove that h a. there are a number of ways to do this but just pick one and write it briefly but clearly