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Let X be a topological space, and let $x_0$ be a point of X. Show that if $\lambda_1$ and $\lambda_2$ in $\pi(X; x_0)$ have the same image under $\pi(X; x_0) \rightarrow [S^1;X]$, then $\lambda_1$ and $\lambda_2$ are conjugate. (Conjugate in the group sense).

I'm confused as to what the images look like in $[S^1; X]$

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    Hint: $\pi_1$ is also defined in terms of loops in $X$ — so what's the difference b/w $\pi_1(X)$ and $[S^1,X]$?..2011-11-19

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