# What is the relationship between various groups

Does the projective linear group of a given dimension share a representation with a group of automorphisms?

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The projective linear group of a vector space is the quotient of the general group by its center, which over a field (or a division ring) is the set (subgroup) of all non-zero scalar matrices...is this what you have in mind? And now, what do you mean by "sharing" a representation? – DonAntonio Feb 1 '13 at 23:14
The projective linear group $PGL(n+1)$ is the group of automorphisms of the projective plane $\mathbb{P}^n$ (as an algebraic variety). – Michael Joyce Feb 1 '13 at 23:25
Or, $PGL_2(\mathbb{C})$ are the holomorphic automorphisms of the Riemann sphere. – Alex Youcis Feb 2 '13 at 0:33
@DonAntonio I am wondering if nxn matrices represent a subgroup of some group of automorphisms besides representing the projective transformations. If that doesn't make any sense, its probably cause I don't know what I am saying :s – Jonathan Fischoff Feb 3 '13 at 19:11
@AlexYoucis do you know a similar result for real projective plane? – Jonathan Fischoff Feb 3 '13 at 19:12