  The notion of entropy is ubiquitous both in natural and social sciences. In
the last two decades, a considerable effort has been devoted to the study of
new entropic forms, which generalize the standard Boltzmann-Gibbs (BG) entropy
and are widely applicable in thermodynamics, quantum mechanics and information
theory. In [23], by extending previous ideas of Shannon [38], [39], Khinchin
proposed an axiomatic definition of the BG entropy, based on four requirements,
nowadays known as the Shannon-Khinchin (SK) axioms.
  The purpose of this paper is twofold. First, we show that there exists an
intrinsic group-theoretical structure behind the notion of entropy. It comes
from the requirement of composability of an entropy with respect to the union
of two statistically independent subsystems, that we propose in an axiomatic
formulation. Second, we show that there exists a simple universal class of
admissible entropies. This class contains many well known examples of entropies
and infinitely many new ones, a priori multi-parametric. Due to its specific
relation with the universal formal group, the new family of entropies
introduced in this work will be called the universal-group entropy. A new
example of multi-parametric entropy is explicitly constructed.
