Is it possible to obtain general relativity as a gauge theory from the general linear group?
The starting point is:
$$ M'=GM $$
where $M',M,G$ are elements of $GL(4,\mathbb{R})$.
I believe the second step is to construct a 'pass-throught' derivative (gauge covariant derivative) such that:
$$ D (GM) = GdM \tag{1} $$
According to https://en.wikipedia.org/wiki/Gauge_covariant_derivative#General_relativity, the covariant derivative of general relativity is :
$$ \nabla_j v^I:=\partial_jv^I +\sum_k \Gamma^i_{jk}v^k \tag{2} $$
How can I obtain (2) starting from (1)?
Finally, if you look at this answer To which extent is general relativity a gauge theory? it seems to suggest that you can, but then the comment section of this answer seems to show the author backtracking on the claim.
