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Supersymmetry is a relation between bosons and fermions. Based on this definition, what is the meaning of "number of supersymmetries"?

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As all other symmetries in physics supersymmetry is also described by some algebra. For example, if you consider spin in quantum mechanics the commutation relations are the algebra of the underlying $SU(2)$ symmetry. The three Pauli matrices are the generators of the symmetry. Now supersymmetry is extended symmetry of spacetime unlike spin which is an internal symmetry. SUSY is an extension of our familiar Poincare symmetry. At the level of algebra one can think of this extended symmetry as adding some more "symmetry generators" that make the full algebra closed. These are the SUSY charges. The number of SUSY charges determines the "the number of supersymmetry".

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