Description of a class of finite groups

The Cayley classification problem, which is to give a complete classification of all groups whose orders are equal to a given natural number n, is solved in two ways. First, it is order fixing and studying non-Abelian groups proceeding from the size of the center or from a normality of a Sylow subgroup or other characteristics of the group. The second direction is to consider the whole class of groups of order n with a certain canonical decomposition of its order. For example, we know that if n is a prime number, there exists a unique group of this order. A classical example of the description of groups of order n = pq, where p and q are different prime numbers, is implemented using Sylow theorems. The problem in the general case has no rational solutions; at present, in connection with this, it has undergone some changes. One of new formulations is as follows: to describe groups of order ap, where a is a factor (in the general case, not prime) such that (a, p) = 1. The author describes a group of order with the condition of normality of its Sylow p-subgroup. Note that the order 23 is the first one that presents the full range of groups. In addition to a cyclic group, which exists for any order, this order is inherent to two Abelian noncyclic groups and two non-Abelian groups.

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