Holonomy element from parallel transport around a loop

Let $\pi:P\to M$ be a principal G-bundle with principal connection $\omega$. Fix a basepoint $x\in M$ and a point $p\in P_x$. Let $\gamma:[0,1]\to M$ be a loop based at $x$, meaning $\gamma(0)=\gamma(1)=x$.

Using parallel transport , $p$ is carried to another point in the same fiber:

Because $P_x$ is a right $G$-torsor, there is a unique element $h_\gamma(p)\in G$ such that

This element is the holonomy element of $\omega$ along $\gamma$ based at $p$.

Changing the basepoint in the fiber conjugates the element: if $p'=p\cdot g$, then

Thus the subgroup $\{h_\gamma(p)\}$ depends on $p$, but its conjugacy class depends only on $x$ and defines the holonomy group at $x$.

Examples

1. If $\omega$ is flat and $M$ is simply connected, then $h_\gamma(p)=e$ for all loops, so the holonomy group is trivial.
2. On the Levi-Civita connection of the round 2-sphere, transporting a tangent frame around a latitude circle yields a nontrivial rotation, giving a nontrivial holonomy element.
3. For an abelian structure group $G$, the conjugation ambiguity disappears, so $h_\gamma(p)$ is independent of the choice of $p\in P_x$.