  The definition of period in finite-state Markov chains can be extended to
regular languages by considering the transitions of DFAs accepting them. For
example, the language $(\Sigma\Sigma)^*$ has period two because the length of a
recursion (cycle) in its DFA must be even. This paper shows that the period of
a regular language appears as a cyclic group within its syntactic monoid.
Specifically, we show that a regular language has period $P$ if and only if its
syntactic monoid is isomorphic to a submonoid of a semidirect product between a
specific finite monoid and the cyclic group of order $P$. Moreover, we explore
the relation between the structure of Markov chains and our result, and apply
this relation to the theory of probabilities of languages. We also discuss the
Krohn-Rhodes decomposition of finite semigroups, which is strongly linked to
our methods.
