most recent 30 from http://mathoverflow.net 2013-05-23T08:59:03Z http://mathoverflow.net/feeds/question/118460 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118460/the-symmetric-group-and-the-field-with-one-element H A Helfgott 2013-01-09T16:08:47Z 2013-01-09T16:53:40Z

<p>I've heard a few times that the symmetric group is an algebraic group over a field with one element, and that the alternating group is quite specifically $SO_n(\mathbb{F}_1)$. This does make a lot of sense intuitively, and actually helps to explain to non-specialists the relations between different things I have done. </p> <p>However, objects over the (non-existent) field with one element aren't just a metaphor - they are objects that can be defined properly, though that may not have happened <em>yet</em>. What is the status for the correspondence between $Alt(n)$ and $SO_n(\mathbb{F}_1)$? Is there really a well-defined homomorphism of some sort, and, if so, are there references where this is worked out?</p>