The symmetry group SU(N) transforms N objects into each other. The operation is map composition.
Gravity as curvature is the topological nature and texture of spacetime. Einstein's theory of gravity is based on local symmetries, exact symmetries, and as such is geometric in nature. Electromgnetism, light, a 5-dimensional phenomenon, must have two generators to parallel the operational field of the quintic - the complex field. This is a reflection into a dimension we are not privy to, involving as it does the notion of 'imaginary time,' imaginary in the sense of the second component of a complex number, and in the sense of nonlinear time.
Electromagnetism is realized as the asymptotic action of a spherically represented gravitational generator -- the square of the quintic solution. Perhaps both lower-ordered generators -- nonlinearity and linearity -- are at each of the dual 'right' orientations, perpendicular to each other as vertical and surface slices, intersecting at the center, or the center line, of gravitational symmetry, forming the gravitational envelope.
The strong force and beta decay are 'below the surface' of 3-dimensional reality. Furthermore, beta decay is the asymptotic phenomenon of 2-D strong force. It is responsible not only for the transmutation of one particle into another, but also for the creation of stable intermediate elements to act as midwives for the fusion of heavy elements, elements like carbon and oxygen. Without beta decay, there would be no carbon-based life forms.
With the first one, gravity, each dimension of spacetime can be considered the identity element of the group in a state of superposition. Gravity is all-pervasive, varying locally with time. The graviton, the hypothesized quanta of gravitational force, acts on all particles and their associated fields, including itself.
With the second, electromagnetism, two dimensions adopt the role of identity element essentially collapsing the ten dimensions of spacetime to five, four that we know of, and a fifth in the imaginary time reflection. This fifth dimension does not just add to the known four, but rather increases the complex geometrically, giving each, considered as spatial only, an orientation that collectively can be represented as the envelope of a spherically apprehended universe operating outside the condition of linear time.
The third, the strong force, the force responsible for the interchange of identities between the neutron and proton by the emission or absorption of a quanta of energy -- the gluon -- perculating up, so to speak, from their constituents -- the quarks -- is two dimensional.
Five dimensions of spacetime, represented by the electromagnetic field, take on the role of the identity element. The ten dimensions of a string-theory universe collapse to two. The five dimensions (group elements) of the 'normal group' ['normal' here may refer to time reversibility, or, to the abelian nature of the transformations] collapse the ten dimensions of spacetime in such a way that the universe is seen as a union of pairs of dimensions -- surfaces -- each represented by its own coset. Beta decay -- the weak force -- is one-dimensional, having as identity all of spacetime. All ten dimensions map to the identity element of the group -- a unit of spacetime.
This is all pure speculation on my part, of course.