A tuple $(A_1,\dots,A_n)$ of subsets of a finite group $G$ is called a

factorizationof $G$ if $G=A_1\cdots A_n$ and $|A_1|\cdots|A_n|=G$.

In Cryptology factorizations of groups are known as *logarithmic signatures*.

A factorization $(A_1,\dots,A_n)$ of a group $G$ is called

$\bullet$ *minimal* if for every $i\le n$ the cardinality $|A_i|$ is prime or equal to 4;

$\bullet$ *prime* if for every $i\le n$ the cardinality $|A_i|$ is prime;

$\bullet$ *cyclic* if for every $i\le n$ the set $A_i$ is *cyclic* in the sense that $A_i=\{a_i^k:0\le k<|A_i|\}$ for some $a_i\in A_i$;

$\bullet$ a *decomposition* if each set $A_i$ is a subgroup of $G$.

Since each group of prime cardinality is cyclic, each prime decomposition is cyclic.

It is easy to see that each cyclic set $A$ can be written as the product $A=A_1\cdots A_n$ of cyclic sets $A_1,\dots,A_n\subset A$ of prime cardinality with $|A|=|A_1|\cdots|A_n|$.

Consequently, for any finite group $G$ we have the implications:

$G$ has a prime decomposition $\Rightarrow$ $G$ has a cyclic decomposition $\Rightarrow$ $G$ has a cyclic factorization $\Rightarrow$ $G$ has a prime factorization $\Rightarrow$ $G$ has a minimal factorization.

For any prime number $p$ and any $n>1$ the cyclic group $C_{p^n}$ has a cyclic decomposition (and hence has a prime factorization) but does not have a prime decomposition.

The 8-element group $Q_8=\{1,-1,i,-i,j,-j,k,-k\}$ of quaternion units has a cyclic factorization but does not have a cyclic decomposition.

For other 3 properties we have 3 (open?) problems:

Problem M.Has every finite group $G$ a minimal factorization?

Problem P.Has every finite group $G$ a prime factorization?

Problem C.Has every finite group $G$ a cyclic factorization?

Problem M is well-known in Cryptography as the problem of existence of a minimal logarithmic signature. Minimal (and cyclic) logarithmic signatures have been constructed in many classical finite groups and also in some sporadic simple groups. By Theorem 2.1 in the Ph.D.Thesis of Singhi, solvable groups and all alternating groups have cyclic factorizations. Moreover, by the proof of Proposition 7.1 in this paper of Magliveras, every alternating group $A_n$ has a cyclic decomposition.

Nonetheless, Problem M is known to be open.

I am interested in the status of Problems P and C.

Have we a counterexample to Problem C?

What is the role of the number 4 in the definition of a minimal factorization?

Is there any example of a finite group for which we know that a minimal factorization exists but a prime (or cyclic) factorization cannot be constructed?