Is group theory used at all in the study of supersymmetry? I have encountered Lie groups in gauge symmetry, but I was wondering if anyone could give me some specific examples of mathematical group theory concepts being used in determining the properties of SUSY particles, if any.
 A: In general, whenever one is describing the symmetries of a theory in physics, one is automatically dealing with group theory, and supersymmetry is not an exceptional case.
The Coleman-Mandula no-go theorem in essence states that any QFT (under certain assumptions) can only realise symmetries being some direct product of the Poincaré group and an internal group. Supersymmetry naturally arises in this context by considering spinor generators. All this is set in the language of group and representation theory.
For the more mathematically inclined, as explained in Supersymmetry and Supergravity, one can arrive at supersymmetry by considering graded Lie algebras, and this extends to how superspace is naturally considered after studying graded vector spaces.
Though, the notion of a superfield and superspace arises directly from group theory too. If we think of the SUSY algebra as one of anti-commuting parameters, we can define a group element,
$$G(x,\theta,\bar\theta) = \exp \left(-ix^mP_m + \theta Q + \bar\theta \bar Q\right)$$
and by interpreting group multiplication as a motion in the parameter space, one can define differential operators $Q$ and $\bar Q$ for left multiplication (as well as for right multiplication). Naturally we arrive at the question: what do these operators act on? The answer is functions on this parameter space - superfields.
In the same way we can build Lorentz scalars, we can build objects transforming suitably under supersymmetry transformations with these superfields (made up of component fields), allowing us to construct Lagrangians, but the whole idea is first founded upon group theory.
