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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.

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- Thread starter patelnjigar
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- #1

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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

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what have you tried?

- #3

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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

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(i) if x is in G, then so is its inverse

(ii) if x,y are in G, then so is xy

- #5

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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

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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

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

- #8

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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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