In physical applications, fiber bundles often come with a preferred group of transformations (usually the symmetry group of the system). The modem attitude of physicists is to regard this group as a fundamental structure which should be implemented from the very beginning enriching bundles with a further structure and defining a new category.
A similar feature appears on manifolds as well: for example, on ℜ^{2} one can restrict to Cartesian coordinates when we regard it just as a vector space endowed with a differentiable structure, but one can allow also translations if the “bigger” affine structure is considered. Moreover, coordinates can be chosen in much bigger sets: for instance one can fix the symplectic form w = dx ∧ dy on ℜ^{2} so that ℜ^{2} is covered by an atlas of canonical coordinates (which include all Cartesian ones). But ℜ^{2} also happens to be identifiable with the cotangent bundle T*ℜ so that we can restrict the previous symplectic atlas to allow only natural fibered coordinates. Finally, ℜ^{2} can be considered as a bare manifold so that general curvilinear coordinates should be allowed accordingly; only if the full (i.e., unrestricted) manifold structure is considered one can use a full maximal atlas. Other choices define instead maximal atlases in suitably restricted subclasses of allowed charts. As any manifold structure is associated with a maximal atlas, geometric bundles are associated to “maximal trivializations”. However, it may happen that one can restrict (or enlarge) the allowed local trivializations, so that the same geometrical bundle can be trivialized just using the appropriate smaller class of local trivializations. In geometrical terms this corresponds, of course, to impose a further structure on the bare bundle. Of course, this newly structured bundle is defined by the same basic ingredients, i.e. the same base manifold M, the same total space B, the same projection π and the same standard fiber F, but it is characterized by a new maximal trivialization where, however, maximal refers now to a smaller set of local trivializations.
Examples are: vector bundles are characterized by linear local trivializations, affine bundles are characterized by affine local trivializations, principal bundles are characterized by left translations on the fiber group. Further examples come from Physics: gauge transformations are used as transition functions for the configuration bundles of any gauge theory. For these reasons we give the following definition of a fiber bundle with structure group.
A fiber bundle with structure group G is given by a sextuple B = (E, M, π; F ;>.., G) such that:
 (E, M, π; F) is a fiber bundle. The structure group G is a Lie group (possibly a discrete one) and λ : G —–> Diff(F) defines a left action of G on the standard fiber F .

There is a family of preferred trivializations {(U_{α}, t_{(α)}}_{α∈I} of B such that the following holds: let the transition functions be gˆ_{(αβ)} : U_{αβ} —–> Diff(F) and let e_{G} be the neutral element of G. ∃ a family of maps g_{(αβ)} : U_{αβ} —–> G such
that, for each x ∈ U_{αβγ} = U_{α} ∩ U_{β} ∩ U_{γ}
g_{(αα)}(x) = e_{G}
g_{(αβ)}(x) = [g_{(βα)}(x)]^{1}
g_{(αβ)}(x) . g_{(βγ)}(x) . g_{(γα)}(x) = e_{G}
and
gˆ_{(αβ)}(x) = λ(g_{(αβ)}(x)) ∈ Diff(F)
The maps g_{(αβ)} : U_{αβ} —–> G, which depend on the trivialization, are said to form a cocycle with values in G. They are called the transition functions with values in G (or also shortly the transition functions). The preferred trivializations will be said to be compatible with the structure. Whenever dealing with fiber bundles with structure group the choice of a compatible trivialization will be implicitly assumed.
Fiber bundles with structure group provide the suitable framework to deal with bundles with a preferred group of transformations. To see this, let us begin by introducing the notion of structure bundle of a fiber bundle with structure group B = (B, M, π; F; x, G).
Let B = (B, M, π; F; x, G) be a bundle with a structure group; let us fix a trivialization {(U_{α}, t_{(α)}}_{α∈I} and denote by g_{(αβ)} : U_{αβ} —–> G its transition functions. By using the canonical left action L : G —–> Diff(G) of G onto itself, let us define gˆ_{(αβ)} : U_{αβ} —–> Diff(G) given by gˆ_{(αβ)}(x) = L (g_{(αβ)}(x)); they obviously satisfy the cocycle properties. Now by constructing a (unique modulo isomorphisms) principal bundle P_{B} = P(B) having G as structure group and g_{(αβ)} as transition functions acting on G by left translation L_{g} : G —> G.
The principal bundle P(B) = (P, M, p; G) constructed above is called the structure bundle of B = (B, M, π; F; λ, G).
Notice that there is no similar canonical way of associating a structure bundle to a geometric bundle B = (B, M, π; F), since in that case the structure group G is at least partially undetermined.
Each automorphism of P(B) naturally acts over B.
Let, in fact, {σ^{(α)}}_{α∈I} be a trivialization of P_{B} together with its transition functions g_{(αβ)} : U_{αβ} —–> G defined by σ^{(β)} = σ^{(α)} . g_{(αβ)}. Then any principal morphism Φ = (Φ, φ) over P_{B} is locally represented by local maps ψ(α) : U_{α} —> G such that
Φ : [x, h]_{α} ↦ [φ_{(α)}(x), ψ_{(α)}(x).h]_{(α)}
Since Φ is a global automorphism of P_{B} for the above local expression, the following property holds true in U_{αβ}.
φ_{(α)}(x) = φ_{(β)}(x) ≡ x’
ψ_{(α)}(x) = g_{(αβ)}(x’) . ψ_{(β)}(x) . g_{(βα)}(x)
By using the family of maps {(φ_{(α)}, ψ_{(α)})} one can thence define a family of global automorphisms of B. In fact, using the trivialization {(U_{α}, t_{(α)}}_{α∈I}, one can define local automorphisms of B given by
Φ^{(α)}_{B} : (x, y) ↦ (φ_{(α)}(x), [λ(ψ_{(α)}(x))](y))
These local maps glue together to give a global automorphism Φ_{B} of the bundle B, due to the fact that g_{(αβ)} are also transition functions of B with respect to its trivialization {(U_{α}, t_{(α)}}_{α∈I}.
In this way B is endowed with a preferred group of transformations, namely the group Aut(P_{B}) of automorphisms of the structure bundle P_{B}, represented on B by means of the canonical action. These transformations are called (generalized) gauge transformations. Vertical gauge transformations, i.e. gauge transformations projecting over the identity, are also called pure gauge transformations.