# $GL(n,G)$ General Linear group of a group $G$?

I know you can take the General linear group of some vector space $$V$$: $$GL(V)$$.

For example, suppose I have a three-sphere, elements of which I might represent as $$SU(2)$$ since they're diffeomorphic. I know that the three sphere admits a set of three linearly independent vector fields, the $$\it{basis}$$ of which can be represented by elements $$SU(2)$$ (or equivalently the $$i,j,j$$ of the quaternions).

So given such a space, how would I go about finding the General Linear group of that group (maybe that's a bad way to word it)?

Or maybe I should say how do I find the General linear group with that structure imposed upon it. How do I find out how that group differs from one on Euclidean 3-space, in a group theoretic manner. (apologies for poor terminology, my background is in physics).

I was hoping to do this for more general cases and groups like the special affine group of some particular space (for example).

• You need to be able to add elements in $G$ to define the product of matrices. You at least need a semiring. – Pedro Tamaroff Dec 23 '18 at 12:56
• As an aside, what makes you think the 'special affine group' is isomorphic to the double cover of the Poincaré group? The dimensions don't seem to match. – Thomas Bakx Dec 23 '18 at 19:45
• @ThomasBakx The special affine group: $$SA(4,R)=SL(4,R)\ltimes R^{1,3}$$ (for Minkowskian space) which is isomorphic to: $$SL(2,C)\ltimes R^{1,3}$$, which is the connected double cover of the Poincare group. I'm pretty sure that's right. I decided to ask about it here: physics.stackexchange.com/questions/450963/… – R. Rankin Dec 30 '18 at 1:23
• @ThomasBakx I could be wrong (though i'd like to know either way) math.stackexchange.com/questions/3048029/… It would seem at the very least, The connected double cover is a subgroup of the SAG? – R. Rankin Dec 30 '18 at 1:30
• @R.Rankin The identity matrix is special unitary, but twice the special matrix is not. Similarly, the sum of two unit quaternions is not, in general, a unit quaternion. – Pedro Tamaroff Dec 30 '18 at 2:01