Associated bundle Given a principal $G$-Bundle $P\rightarrow X$ and if we let $G$ act on itself by multiplication (denote this action by $\rho$) we obtain an associated bundle $P\times_{\rho} G=(P\times G)/\sim$ where $(p,f)\sim(gp, gf)$ with fibers homeomorphic to $G$. It is easy to check that this bundle is G-principal; the G-action is given by $g[p,f]=[p,fg]$. 
If we change the action of $G$ on itself to conjugation (denote this action by $\phi$) we can construct an associated bundle $P \times_{\phi} G$ in the same way. However, I want to show that this is not a principal bundle by showing that the fibers are not only homeomorphic to $G$ but they are groups isomorphic to $G$. (This would imply that the bundle has a section and if it were principal it would be trivial and there are examples of non trivial associated conjugation bundles). 
How can I show that each fiber of $P \times_{\phi} G$ is a fiber bundle of groups? I would probably have to use the fact that conjugation by a fixed element is a group isomorphism, since in the case of multiplication, which is not a group isomorphism, the obtained bundle is principal.
 A: I'm not sure exactly what you mean by "fibers are not only homeomorphic to $G$ but they are groups isomorphic to $G$" - what opeartion are you putting on the fibers before asking this question?
On the other hand, I think I can prove there is a section without this part.
Define a map $X\rightarrow P\times_\phi G$ by $x\rightarrow [(p,e)]$ where $p$ is any element of $P$ with $\pi(p) = x$ and $e\in G$ the identity element.
I first claim this map is well defined.  For if $\pi(p) = \pi(p')$, then there is an element $g\in G$ with $gp = p'$.  For this $g$, we then have $[(p,e)] = [g*(p,e)] = [(gp, geg^{-1}] = [(p',e)]$, so the map is well defined.
To see it's continuous, note that this can be checked locally.  If $U\subseteq X$ with both the principal and associated bundles trivial over $U$, then, when restricted to $U$, the map from $X$ to $P\times_\phi G$ factors as $X\rightarrow U\times G\times G\rightarrow P\times G\rightarrow P\times_\phi G$ where the first map sends $x$ to $(x,e,e)$ and the second map sends $U\times G$ to $\pi^{-1}(U)\subseteq P$ via a trivialization and the third map is the natural projection.  Since all of these maps are continuous, our section factors (locally) as a composition of continuous functions, hence is continuous itself.
