# an equation in a component of identity in a lie group

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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It is not clear where this question comes from, can you give more context? One could imagine trying to trace solutions by moving $y$ around, starting at $y=e$ (where by the way there are in general more solutions than just $x=e$), which is an approach that would require a connected group. But it is not at all clear that this is actually a very promising angle of attack. – Marc van Leeuwen Sep 24 '12 at 8:27
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? – miosaki Sep 24 '12 at 10:34

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.