I am currently writing a final project on simple groups and my professor said I should focus on "the most important simple group", $PSL(2,F)$. I have also read in Aluffi's algebra book that $PSL(2,F)$ is a "candidate for the most important group in mathematics". I picked up Rotman's 'An Introduction to the Theory of Groups', which has some nice results on $PSL(2,F)$. What I am failing to see is what makes this group so important and what influence it has on the whole of group theory/algebra. I certainly understand why simple groups are important, but I fail to see why this particular simple group is so important.
Does anyone have any references where I can read more about what specifically makes this group so darn important, or have any specifics on the importance of it? Thanks!