jimbothecannon28
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Hi There,

Have been learning OCR FP3 Group Theory and all has been going well however have just started isomorphic groups and getting a little confused. I have the characteristics of isomorphic groups in my notes but questions which I have been set for homework (Such as the following) has left me a bit stuck and am in need of a pointer! Thanks in advance

"Use the function f(x) = exp(x) to show that (R,+) is isomorphic to (R+, x)

Where R = The set of real numbers (couldn't find symbol).
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Indeterminate
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(Original post by jimbothecannon28)
Hi There,

Have been learning OCR FP3 Group Theory and all has been going well however have just started isomorphic groups and getting a little confused. I have the characteristics of isomorphic groups in my notes but questions which I have been set for homework (Such as the following) has left me a bit stuck and am in need of a pointer! Thanks in advance

"Use the function f(x) = exp(x) to show that (R,+) is isomorphic to (R+, x)

Where R = The set of real numbers (couldn't find symbol).
Well, this question needs you to know the following:

A group (G,+) is isomorphic to another group (H,x) if there is a one-one map

f: G \rightarrow H

such that f(x+y) = f(x)f(y) for all x and y in G.
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morgan8002
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(Original post by jimbothecannon28)
Where R = The set of real numbers (couldn't find symbol).
[tex]\mathbb{R}[/tex]\rightarrow\mathbb{R}
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jimbothecannon28
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(Original post by Indeterminate)
Well, this question needs you to know the following:A group (G,+) is isomorphic to another group (H,x) if there is a one-one map f: G \rightarrow Hsuch that f(x+y) = f(x)f(y) for all x and y in G.
Thanks for the reply OK so I know the conditions for Isomorphism. I'm not sure if we have gone over this in enough detail in class. We learnt the condition that f(xy) = f(x)f(y) for isomorphism and I understand that the operators are different in each group which is why we are saying something different. Or f(x+y)=f(x)f(y) but ensure from here...
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jimbothecannon28
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(Original post by Indeterminate)
Well, this question needs you to know the following:

A group (G,+) is isomorphic to another group (H,x) if there is a one-one map

f: G \rightarrow H

such that f(x+y) = f(x)f(y) for all x and y in G.
Or are you literally just able to say that for all x and y within G

ex+y =exey
(ex)(ey)=exey

?
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Indeterminate
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(Original post by jimbothecannon28)
Thanks for the reply OK so I know the conditions for Isomorphism. I'm not sure if we have gone over this in enough detail in class. We learnt the condition that f(xy) = f(x)f(y) for isomorphism and I understand that the operators are different in each group which is why we are saying something different. Or f(x+y)=f(x)f(y) but ensure from here...
You just need to remember that the LHS of the condition corresponds to the operator on G, and the RHS corresponds to the operator on H.

Ok, so now all you have to do is show that f is one-one and that the condition is satisfied
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