## Reflections and Reports of Representation Theory – Spring 2015 – 1

I’m sitting in a course on representation theory this semester and though it will primarily deal with finite groups I’m extremely excited!  The professor is very good, very clear and motivates material very well.  The past few days has been a review of basic group theory but today a few nice items came up I wanted to share.

I’ve known about the first isomorphism theorem for a while but I saw a lovely example of it today.  Take the general linear group over an arbitrary field F then the determinant is a surjective homomorphism $det: GL_n(F)\to F^*$ where F* is the field without it’s ‘addition’ identity (i.e. zero).  Then the kernel of this map  are those matrices with determinant a unit in F, so $SL_n(F)$!  While a quotient like $GL_n/SL_n$ certainly taxes my mind, we know from the first isomorphism theorem that this is isomorphic to F*.

I’ll try to keep this blog in the loop of this class.  Also I’m trying to form a reading group on campus on the Foundations of Mathematics by Kunen so with any luck I will have some posts on that as well.

I keep putting off learning group theory proper, and instead often work with just specific matrix representations of groups used in particle physics…you’re forcing my hand here! Can you give me an example? I’m guessing $\latex U(n) \rightarrow SU(n)$ in one, but I need to match up terms/definitions…
That is $U(n) \rightarrow SU(n)$