$\pi:E \longrightarrow M$ vector bundle with connection $\nabla$ and structure group $G\leq GL(V)$. I read three definition of G-connection:
1) The holonomy group at $p$ are the parallel transports $L:E_p\longrightarrow E_p$ along loops at $p$. $\nabla$ is a $G$-connection if its holonomy group is "contained" in $G$. Poor's book "Differential geometric structures" 2.48 page 71
2) Wikipedia "A G-connection on E is an Ehresmann connection such that the parallel transport map τ : Fx → Fx′ is given by a G-transformation of the fibers"
3) If every parallel transport $L:E_p\longrightarrow E_q$ is of the form, $L=h_1\circ h_0^{-1}$, where $h_0:V \longrightarrow E_p$, $h_1:V \longrightarrow E_q$ are admissible diffeomorphism. An article about Ehresmann connections
I'm not sure about the meaning of the definitions, what I understand is the following:
1) The holonomy group is contained in "G" means: for any bundle charts $\psi$, $\varphi$, $\psi|_{E_p}\circ L \circ \varphi|_{E_p}^{-1}:V \longrightarrow V \; \in G$
2) If for any two bundle charts $\psi$, $\varphi$ around $q$ and $p$ and every parallel transport between the points $\psi|_{E_q} \circ L\circ \varphi|_{E_p}^{-1} \in G$.
3) $h_0$, $h_1$ are admissible diffeomorphism if exists bundle charts $\psi$, $\varphi$, $h_0=\varphi|_{E_p}^{-1}:V \longrightarrow E_p$, $h_1=\psi|_{E_q}^{-1}$. Then $\psi|_{E_q} \circ L\circ \varphi|_{E_p}^{-1}=id_V$.
It's clear that $3) \Leftrightarrow 2) \; \Rightarrow 1)$. But I don't see why $1)$ implies $2)$ and $3)$.