Recent results on syntactic groups of codes

Finite permutation groups can be considered as building blocks of finite transformation monoids. Indeed, by the Krohn-Rhodes Theorem, any finite transformation monoid divides a semidirect product of permutation groups and three element transformation monoids. Given a transformation monoid M on a finite set, defined by a finite set of generators, it is in general non-trivial to obtain a description of the groups contained in M . For example, there is no simple algorithm to check that all these groups are trivial (see [17]). This paper, which is of expository nature, contains a new presentation of the techniques that allow to compute the groups contained in a transformation monoid. We use this technique to present a survey on some recent results that allow to describe the groups contained in transformation monoids. These transformation monoids arise as the transition monoid M of the minimal automaton recognizing the submonoid generated by a prefix code X . The maximal groups contained in M are called the syntactic groups of X . We treat two cases. In the first case, we discuss the problem of building bifix codes with the minimal possible number of elements having a given syntactic group. In the second one, we show that, for a particular class of prefix codes, all proper syntactic groups are cyclic.