Help with Abstract Algebra: Show ac=b, da=b w/Hint

  • #1
Please I need your help for that qustion and how do slove that qustion's problem. can you help me for slove for that? Pleasee

Let G be any group, and let a, b ∈ G. Show that there are c, d ∈ G such that ac = b and da = b. Hint: you have to give an explicit definition for c and d in terms of a and b.
 
  • #2
what have you tried?
 
  • #3
I have tried set up matrices for that but not work. I don't know how to slove for this problem...


Let G be any group, and let a, b ∈ G. Show that there are c, d ∈ G such that ac = b and da = b. Hint: you have to give an explicit definition for c and d in terms of a and b.
 
  • #4
Matrices?? All you need is:
(i) if x is in G, then so is its inverse
(ii) if x,y are in G, then so is xy
 
  • #5
i m tried work on this..

GIVE a, b ∈ G
SHOW c, d ∈ G ?

ac=b da=b

ac=b ---> proof: a (a^(-1)b)=b (IS THAT RIGHT? I THINK SO AND THAT'S RIGHT)

da=b ---> proof: ?
 
  • #6
if i want to say about ac=b

proof: a (a^(-1)b)=b
uniqueness
if ac=ad=b
then c=d (left Cancellation)

How do I slove for da=b?? I need for that..
 
  • #7
Same idea: let d = ba^(-1).
 
  • #8
Same idea: let d = ba^(-1). ?? I seem that not enough

ac=b
a (a^(-1)b)=b, if ac=ad=b, then c=d (left cancellation).. I m sure that's right.



da=b
d=ba(a^(-1)), if ad=ac=b, then d=c (right cancellation) is that right?
 

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