1. Introduction

Finite p-groups are ideal objects for combinatorial and cohomological investigations. Some of its basic properties were proved by Sylow, Frobenius, and Burnside. But, namely, Philip Hall (1904-1982) laid the foundations of modern p-group theory (Ber. Y [10]). Normally, Blackburn also made very outstanding achievements after Hall. A common method of investigation in algebra is to break up a complex structure into simpler substructures. The hope is that by repeated application of this procedure, one will eventually arrive at structures that are easy to understand. It may then be possible, in some sense, to synthesis these substructures, so as to reconstruct the original one. While it is rare for the procedure just described to be brought to such a perfect state of completion, the analytic synthetic method can yield valuable information and suggest new concepts. Group theory in general, considers how far this can be used. (Kurosh A.G., 1960) [12]. In this work we examine the basic properties of any p-group which put them in categories and hence define the individual and general characterisations.

The hall enumeration principle [10] paves way for counting the subgroups of given structures.

Definition: Let p be a prime. A group G is said to be a p-group if the order of every element of G is a power of p. A group which is finite is a p-group if and only if the order of G is a power of p.

Remark: (Kuk A.) [13] Let G be a group of order m, and let p be a prime. If is the highest power of p which divides m, where d is an integer, then the subgroup of order is called a sylow p-subgroup of G.

Definition: Elementary Abelian Group: Let G be a p-group. If G is abelian and all its elements (with the exception of the identity) have order p; then, the order of G, for some . In this case, G is a direct product of m groups of order p. We then say that G is an elementary abelian p-group.

Remark: Two elementary abelian p-groups of the same order are isomorphic [Ber.Y].

We denote such group by and call it the elementary abelian p-group.

Let be the elementary abelian group of order (see [7], [8], [9]).

Set

Let . Then the following assertions are true (see [10], [11]):

Main Result

The number of subgroups of order in a finite p-group of order is normally distributed with respect to the positive integer .

Hence, by defining a probability function on G, assuming that the area under the graph is a unit, it is possible to estimate the number of subgroups of G.

This is one of the main objectives of studying the modern finite p-groups.

2. Proof of Results

Definition: Suppose that G is any group such that is the identity, then and are both subgroups of G called the trivial subgroups of G.

By computation, from (a) above, considering simple even and odd cases (for m = 4 & m = 5 respectfully), we have that:

(1) For m = 4, the trivial subgroup of G (the identity).

(2) For m = 5

Note here that for m = 4,

And, for m = 5, we have that

Given that, for

where

for

Consider the case:

Observe that as n increases from the left (from below), increases to a maximum value at the middle.

Also, as n decreases from the right (from above), increases in value to the same maximum at the middle. This is seen from the simple computation that:

or for odd m.

Hence, we have the following:

Proposition: The limit of as n increases from below is maximum at for even m and at or for odd m and we write:

Moreover,

and

This is a symmetric property.

In particular, when m is odd, we ague that is normally distributed with n, leading to the normal distribution curve.

Where the area under the curve indicates the number of subgroups of order