Hi! I was just going through this script on Lie groups: http://www.mit.edu/~ssam/repthy.pdf(adsbygoogle = window.adsbygoogle || []).push({});

At one point the following is said:

(see attachment)

I've spent multiple hours trying to figure out why this is a group homomorphism. Sure, once you know the theorem is correct, this follows. But without knowing this, I can't figure out why it should be.

I want to see that:

[itex]\varphi_y(x_1x_2) = \varphi_y(x_1)\varphi_y(x_2) \Leftrightarrow x_1x_2yx_2^{-1}x_1^{-1}y^{-1} = x_1yx_1^{-1}y^{-1}x_2yx_2^{-1}y^{-1}[/itex]

If I write [itex]x_1y = \bar y_1x_1, x_2y = \bar y_2x_2[/itex], then the above equation is equivalent to

[itex]x_1\bar y_2x_1^{-1} = \bar y_1y^{-1}\bar y_2[/itex]

but I don't get any further from here.

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# Xyx^-1y^-1 a Lie group homomorphism?

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