# Theory of gr.group: Are 4 important in the minimum logarithmic signature problem?

One of the problems open in finite groups (which has applications in cryptology) is the problem of the existence of a minimum logarithmic signature for each finite group, see this Ph.D. Thesis for more information.

This problem asks if the following conjecture is true.

Conjecture 1. Each finite group $$G$$ It can be written as the product. $$G = A_1 cdots A_n$$ of subsets $$A_1, points, A_n subset G$$ such that $$| G | = | A_1 | cdots | A_n |$$ and the cardinality of each set. $$A_i$$ is a prime or equal to 4.

What is the role of number 4 in this conjecture? What is known about the next strongest (and most natural) modification of Conjecture 1?

Conjecture 1 & # 39; Each finite group $$G$$ It can be written as the product. $$G = A_1 cdots A_n$$ of subsets $$A_1, points, A_n subset G$$ of cardinality first such that $$| G | = | A_1 | cdots | A_n |$$.

Is there a counter-example with conjecture 1 & # 39 ;?