# Coset spaces of a lie group and fiber bundles

I am reading Steenrod. He writes:

Another example of a bundle is a Lie group $B$ operating as a transitive group of transformations on manifold $X$. The projection is defined by selecting a point $x_0 \in X$ and defining $p(b)=b(x_0)$. If $Y$ is the subgroup that fixes $x_0$ then the fibers are just the left cosets. There are many natural correspondence $Y \rightarrow Y_x$ any $b \in Y_x$ defines one by $y \rightarrow by$. However, any two such maps given by $b$ and $b'$ differ by the left translation of $Y$ corresponding to $b^{-1} b'$. (here is what I do not get:) Thus, the group $G$ of bundle coincides with the fiber $Y$ and acts on $Y$ by left translation. What does this statement mean? Earlier in the book it seems Steenrod is interested in maps from the fibers into $Y$ and then the maps of Y onto itself that are the difference of those two maps. But, here $b^{-1} b'$ is not a map of $Y$ onto itself. This does not seem to have any significance in terms of $Y$ being the structure group.

I think I understand it. The structure group is not the group which acts on fibers and sends them to each other. But, it is the homemorphism of $Y$ onto itself. "However, any two such maps given by $b$ and $b'$ differ by the left translation of $Y$ corresponding to $b^{-1} b'$." has no significance other than giving us an explicit transition maps for fibers, not related directly to the structure group. This is related to this topic as well, I think: Proof of $G\rightarrow G/H$ is a Principal H bundle