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Essentially, I'm trying to prove that when computing the tangent space for a group that there's nothing special about considering it at only the identity. Namely, there is an isomorphism of vector spaces from the tangent space at the identity to the tangent space at any other element of the group.

The method that I'm being guided along is the following:

First, I'm supposed to prove that if I have any smooth curve through an element g, it is of the form gA(t) where A(t) is a smooth curve through the identity. (Proving this is where I'm failing.) Naturally, once I've done this, the isomorphism is there for the isomorphing...

Apologies for any ambiguities in the wording, I'm not exactly firing on all cylinders...

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    What's confusing about the hint? A tangent vector is an equivalence class of smooth curves, and you've just been told how to identity smooth curves through $g$ with smooth curves through the identity.2012-03-29
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    Would it help to say: for $C(t)$ a curve through $g$, $g^{-1}C(t)$ is a curve through the identity. I'm not exactly clear as to where you are having trouble . . .2012-03-29
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    @ZhenLin I'm having trouble proving that every curve through g is of the form gA(t). Namely, for a given curve B(t) through g, why is B(t) always of the form gA(t)?2012-03-29
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    Because I can multiply by $g^{-1}$ pointwise.2012-03-29
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    Oops. No need to shout...2012-03-29
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    To spell out my argument more fully, take a curve $B(t)$ through $g$. Then $g^{-1}B(t)$ goes through the identity. Set $A(t)=g^{-1}B(t)$. Then $A(t)$ is a curve through the identity and $gA(t)=B(t)$.2012-03-29

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