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could any one help me how to solve :

prove that there exist solution for the equation $x^2=y$ in identity component of a lie group. I dont know how to start this one, what is the specia; about component of identity? well for $y=e$ we get $x=e$

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    component of identity is an Normal Subgroup of the Lie Group, so can we just try to see in a group this kind of equation has solution or not?2012-09-24

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The result is false: the matrix $ y=\begin{pmatrix}-1&0\\0&-2\end{pmatrix} $ lies in the identity component of $G=\mathbf{GL}(2,\mathbf R)$ (which is the subset of matrices with positive determinant), but the equation $x^2=y$ for $x\in G$ has no solution, as can be checked by writing the equations for the matrix coefficients explicitly and showing the absence of real solutions.

See also this question relating this equation to the image of the exponential map, and this other question.