Currently, I'm reading the script of a Global Analysis lecture at my university.
There, we look at a real vector-bundle $(E,\pi,M)$ of rank $k$ with a given connection $\nabla$ and define the holonomy group at the point $p\in M$ as $$\text{Hol}_p(\nabla):=\{P_\gamma:\gamma\text{ is a closed curve with basis }p\}.$$ Here, $\gamma:[0,1]\rightarrow M$ is continuous with $\gamma(0)=\gamma(1)=p$ and $P_\gamma$ is the unique parallel transport along $\gamma$.
We have shown beforehand that $P_\gamma$ is an isomorphism and that there is a certain mulitplicative action, i.e. $P_{\gamma\gamma\prime}=P_\gamma\circ P_{\gamma\prime}$.
Hence, $\text{Hol}_p(\nabla)$ can be looked at as a subgroup of $\text{GL}(k,\mathbb{R})$.
And suddenly, my problem appears: We set $G:=\text{GL}(k,\mathbb{R})/$~, call this the holonomy group of $E$ w.r.t. $\nabla$ and state $\text{Hol}_p(\nabla)\cong G$.
But what's the equivalence relation? I thought fruitlessly, and Wikipedia doesn't state it this way.