Can the structure group be different than the Lie group fiber of a principal bundle?

Question

Let $\pi: P \rightarrow M$ be a principal $G$-bundle whose fibers are a Lie group $G$. In Hamilton's Mathematical Gauge Theory he states that

The group $G$ is called the structure group of the principal bundle.

I was left thinking that the term structure group is another way to refer to the fiber of a principal bundle.

However, I have also come across other resources (such as this question: What is the structure group of the tangent bundle?) that gives a more general definition where the structure group can be completely different than the fiber. From what I have understood of their definition, let $\pi: E \rightarrow M$ be a fiber bundle with fiber $F$. If $u$ is in the domain of two trivializations $\phi_1: E \rightarrow M \times F$ and $\phi_2: E \rightarrow M \times F$, with images $\phi_1(u) = (x, f_1)$ and $\phi_2(u) = (x, f_2)$, then the structure group $G$ of the fiber bundle is a group whose group action relates these two images, namely $f_1 = g \cdot f_2$ for some $g \in G$.

I am now a little confused on what the structure group is exactly and how it relates to the fiber of a fiber bundle, especially in the context of principal bundles. In the case where we have a principal $G$-bundle is the structure group always the same as the fiber (and hence the Lie group) $G$ or can it be generalized to another group?

Answer 1

What’s in the link is actually not a more general definition, it is more restrictive. Here’s what Hamilton writes; immediately after giving the definition of a principal bundle, he says

Remark 4.1.3: The classic references [133] (Steenrod) and [81] (Husemoller) use the term fibre bundle in a more restrictive sense; see Remark 4.1.15.

and remark 4.1.15 says the following:

Remark 4.1.15: Some references, such as [133] and [81], use the term fibre bundle more restrictively. If the topological definition in these books is transferred to a smooth setting, the definition amounts to assuming that the transition functions of a bundle atlas are smooth maps to a Lie group $G$, acting smoothly as a transformation group on the fibre $F$, instead of maps to the full diffeomorphism group $\text{Diff}(F)$ of the fibre: \begin{align} \phi_{ji}:U_i\cap U_j&\to G\\ x&\mapsto \phi_{jx}\circ\phi_{ix}^{-1}. \end{align} Equivalently, a fibre bundle is with this definition always an associated bundle in the sense of Remark 4.7.8.

So, fiber bundle as defined by Hamilton (and also other authors) does not come equipped with a group, whereas in the other references of Steenrod and Husemoller (let’s just take Husemoller for example) the definition is as the associated fiber bundle:

Definition.[Husemoller, Definition of Fiber Bundle]

Let $\xi=(X,p,B)$ be a principal $G$-bundle and let $F$ be a left $G$-space. The relation $(x,y)s=(xs,s^{-1}y)$ defines a right $G$-space structure on $X\times F$. Let $X_F$ denote the quotient space $(X\times F)/G$ and $p_F$ the factorization of $X\times F\to X\to B$ by the projection $X\times F\to X_F$. With this notation, the bundle $(X_F,p_F,B)$, denoted $\xi[F]$, is called the fiber bundle over $B$ with fiber $F$ (viewed as a $G$-space) and associated principal bundle $\xi$. The group $G$ is called the structure group of the fiber bundle $\xi[F]$.

Reiterating, note that although in this source one calls $G$ “the structure group of $\xi[F]$”, the very definition of fiber bundle is restrictive, in the sense that a fiber bundle is defined to be an associated bundle of a principal bundle (whereas Hamilton requires it to be a triple $(X,p,B)$ where $p$ is surjective and smooth/continuous and satisfy a local-triviality condition… other names for this object include locally-trivial (smooth/continuous) bundle).