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 ;?