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I have a question about the projective general linear group. How does one realize it as a matrix group? Specifically, what is an embedding of $PGL_n \Bbb C \to GL_k \Bbb C$ for some $k$? In this case, $PGL_n \Bbb C$ is defined as the quotient of $GL_n \Bbb C$ by its center.

Thank you.

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The adjoint representation of $PGL_n\mathbb C$ on its Lie algebra is faithful, since the group is connected and has a trivial center. From that you get an embedding in a matrix group.

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    Actually, I believe I figured it out. Since the group is connected and has trivial center, then it is equal to its own adjoint form. Since the image of the adjoint representation is the adjoint form, the kernel must be trivial, giving us our faithful representation. Thanks again Mariano!2011-10-12