UNE application de la theorie des groupes

Received: 02-Jan-2024, Manuscript No. puljpam-24-6987; Editor assigned: 03-Jan-2024, Pre QC No. puljpam-24-6987(PQ); Accepted Date: Jan 29, 2024; Reviewed: 05-Jan-2024 QC No. puljpam-24-6987(Q); Revised: 07-Jan-2024, Manuscript No. puljpam-24-6987(Q); Published: 31-Jul-2024, DOI: 10.37532/2752-8081.24.8(4).01-02

Citation: Strainchamps D. UNE application de la theorie des groupes. J Pure Appl Math. 2024; 8(4):01-02.

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Abstract

this article present a new conjecture that define a method to calculate a sort of integral numeric in using the group theory.

Key Words

Groups; Surjection; Polynomial sum; Logic

Introduction

Let G_b a group of order b and G_b = {g₀,g₁,.,g_b-1} where g₀ is the neutral element of G.

The purpose is to made a surjection, using the group G, between and the set of integers {0, 1, 2, . . . , b − 1} and to do a conjecture with it [1].

In this introduction I will use an example to illustrate this surjection with G = ℤ/4ℤ

Let n∈ ℕ . We convert n in base b = 4. So the integer is equal to:

where any ci is an element of the set {0, 1, 2, 3} and where:

We decide now ∀i<=α that all c_i equals to 0 are replaced by g₀ and the 1 replaced by g₁ and so on respectively the 2 replaced by g₂ and the 3 replaced by g₃.

We can name this new value

And after we made the surjection f with:

This sum is made with the + that is the internal law of G.

Remark 1.1.

If the internal law of G is x, we can do a product.

And so all f(n) are element of G, here in your example ℤ/4ℤ

Finally we associate all the f(n) equals g₀ to the integer 0 and so on in the same order for all members of G with an bijection of identification that we name g

In your example of G = ℤ/4ℤ and all G = ℤ/bℤ we can evidently view that:

Lemma 1.2.

G = ℤ/bℤ also

Remark 1.3.

We have used G = ℤ/4ℤ but it’s clearly evident that the surjection g o f can be defined with the same process with any group G (Figure 1,2).

Figure 1: 0-periode.

Figure 2: Other-periode.

Main Property of the Surjection

In this section we will proove that there is b periode of b^b-1 elements in the results of the surjection g o f

Theorem 3.1.

If we define the b periods with a group G of order b and if we assign as in the section above to all element of the b periods, one element of the natural element {0, 1, 2, ·, b−1} with the surjection g o f then ∀ polynom P of degree < b we have this equality

Proof.

Not yet perfomed but this conjecture do a definition of an integral numeric

Annexes

This script calculate a sum with all their terms in 43 s or calculate the same with 1 term over 6 in 6 s

References

1. Armstrong MA. Groups and Symmetry. Springer-Verlag New York Inc. 1988.

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