  Variable-length codes are the bases of the free submonoids of a free monoid.
There are some important longstanding open questions about the structure of
finite maximal codes, namely the factorization conjecture and the triangle
conjecture, proposed by Perrin and Sch\"{u}tzemberger. The latter concerns
finite codes $Y$ which are subsets of $a^* B a^*$, where $a$ is a letter and
$B$ is an alphabet not containing $a$. A structural property of finite maximal
codes has recently been shown by Zhang and Shum. It exhibits a relationship
between finite maximal codes and factorizations of cyclic groups. With the aim
of highlighting the links between this result and other older ones on maximal
and factorizing codes, we give a simpler and a new proof of this result. As a
consequence, we prove that for any finite maximal code $X \subseteq (B \cup \{a
\})^*$ containing the word $a^{pq}$, where $p,q$ are prime numbers, $X \cap a^*
B a^*$ satisfies the triangle conjecture. Let $n$ be a positive integer that is
a product of at most two prime numbers. We also prove that it is decidable
whether a finite code $Y \cup a^{n} \subseteq a^* B a^* \cup a^*$ is included
in a finite maximal code and that, if this holds, $Y \cup a^{n}$ is included in
a code that also satisfies the factorization conjecture.
