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

[patelnjigar]

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.

 

[mathwonk]

what have you tried?

 

[patelnjigar]

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.

 

[morphism]

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

 

[patelnjigar]

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: ?

 

[patelnjigar]

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..

 

[morphism]

Same idea: let d = ba^(-1).

 

[patelnjigar]

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?