quantum field theory - Why do we want supersymmetry transformations to form a group?

I am getting introduced into supersymmetry reading Ryder "Quantum Field Theory". I have taken an introductory course on QFT last semester so I am far from being an expert. Sorry if this is a silly question.

Now I have a question: why do we want that the transformations we use to exploit the symmetries of theories, such as Poincaré transformations, to form a group?

I ask this question because in Ryder chapter 11 page 448 he introduces two auxiliary fields $F$ and $G$ in order that SUSY transformations to form a group. So it seems important that the transformations to form a group, otherwise we are doing things wrong... Why?

Answer

Consider three observers Alice, Bob and Charles. They have different views of the world on account of them being moved in some way relative to one another. For example Bob might be rotated with respect to Alice whilst Charles might be boosted with respect to Bob. Alice encodes her view of the world by an element $a$ belonging to some set of "states" $A$. So, $a\in A$ is Alice's current state of the world as she measures it. Similarly $b\in B$ is Bob's state of the world as he measures it and Charles sees the world as $c\in C$. Now, since Alice, Bob and Charles are students of physics and not English literature, they believe that they can come up with a consistent description of the world. So, Alice's set of states $A$ and Bob's set of states $B$ must be related by some map $T_{g_{1}}$ such that $T_{g_{1}} : A\rightarrow B$. Here $g_{1}$ is just a label that picks the particular map. Similarly the sets of states of Bob and Charles are related by another map $T_{g_{2}}:B\rightarrow C$. There must also be a map connecting the states of Alice and Charles $T_{g_{3}}:A\rightarrow C$. So, there are two ways of relating the sets of Alice and Charles. We can go $C=T_{g_{2}}B=T_{g_{2}}T_{g_{1}}A$ and $C=T_{g_{3}}A$. Since they believe they can achieve a consistent description of the world, then $T_{g_{3}}=T_{g_{2}}T_{g_{1}}$. This relation suggests that the maps $T_{g}$ are a group representation. This is confirmed by some further investigation. Suppose Bob is not moved with respect to Alice. Then Bob and Alice are the same observer and $A=B$ and so, in this case $T_{g_{1}}$ is the identity map. So, we know that there is an identity. Furthermore, it must be possible to invert the map to get $A=T_{g_{1}'}B$ so every map has an inverse. So, we've got all the rules for a group representation. So, the answer to the question, is that the transformations form a group representation because the physicists believe there is a consistent description of the world for all observers.

Edit:

In a more general way, the observers could be replaced by the freedom in physical description. Alice's set of states $A$ could have some coordinates to label each $a\in A$. The particular coordinates chosen should not be important as each coordinate system should be equivalent. The various coordinate systems would also form a representation of a group.

Regarding the supersymmetry transformations. The physics is described by some fields. The values which the fields can take correspond to Alice's set $A$ in my original explanation. The simplest supersymmetric field theory is the massless, non-interacting Wess-Zumino model. This has a scalar field $\phi$ and a chiral fermion field $\psi^{i}$ with components $i=1,2$. I've only studied supersymmetry a little in order to try to decide if it is well-motivated or not. My understanding is that the scalar field and the chiral field are to be regarded as coordinates of a single object which describes the physics. The supersymmetry transformation is a rotation of these coordinates. The small rotation causes the scalar to change by $\delta \phi$ and the chiral spinor to change by $\delta\psi^{i}$. The physics should be the same regardless of how we choose to rotate the coordinates. This means that the Lagrangian of the theory should be invariant under the rotation up to a total derivative. In the supersymmetric theory, the condition that the rotations form a group $T_{g_{3}}=T_{g_{2}}T_{g_{1}}$ has to be studied for infinitesimal rotations for the sake of simplicity. For infinitesimal rotations this condition becomes the requirement that the commutator of two small rotations is another small rotation (closure). It turns out that the supersymmetry transformations are not closed under the commutator unless the fields are on-shell. However, by inventing an auxiliary scalar field $F$ with trivial dynamics and with a different transformation rule to the original scalar field, the supersymmetry transformations close off shell - they form a group. So we end up with an object with coordinates $\phi,\psi^{i},F$ which describes the physics. The physics should not depend on the how we rotate these coordinates. These rotations are the supersymmetry transformations.

In summary, we have some set $A$ which describes the physics. We have to have some coordinates to label states $a\in A$. There must be a freedom to describe the physics with different coordinates. The transformations (rotations) between different coordinates must be a group.