Exploring Finite Group Theory: Finding the Upper Bound of Groups of Order

In summary: So, in the proof of the theorem, they are using this property of finite sets to show that there is an upper bound on the number of different groups of order $n$. In summary, in the context of group theory, a theorem states that for a given positive integer $n$, there exist finitely different types of groups of order $n$. The proof of this statement involves defining a map from $G\times G$ to a set $X$ with $n$ elements and using the property that there are $n^{n^{2}}$ different mappings from $G\times G$ to $X$ to show that there is an upper bound on the number of different groups of order $n$.
  • #1
pauloromero1983
2
0
In the context of group theory, there's a theorem that states that for a given positive integer \(n\) there exist finitely different types of groups of order \(n\). Notice that the theorem doesn´t say anything of how many groups there are, only states that such groups exist. In the proof of this statement, they define a map \(f:G\times G \rightarrow X\) where \(X\) is a set with \(n\) elements. Defining a group structure in the same map by means of the product rule \(f(g_{1})f(g_{2})=f(g_{1}g_{2})\), where \(g_{1}, g_{2}\) belong to \(G\) they arrive to the following conclusion: there's an upper bound on the number of different groups of order \(n\), namely: \(n^{n^{2}}\)

My question is how to arrive to such conclusion. I am aware that, for every ordered pair of \(G\times G\) there's \(n\) images (since \(X\) was assumed to have \(n\) elements). For a concrete example, let be \(G\) a group of 2 elements. Then, there are 4 ordered pairs. Each pair has 2 images, so the total number of maps would be 4*2=8. However, by use of the relation \(n^{n^{2}}\) we get \(2^{2^{2}}=16\), i.e, there are 16 different maps, not 8. I am missing something here, but I don't know what exactly what the error is.
 
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  • #2
Hi pauloromero1983,

In the example you gave for each element of $G\times G$ there are 2 choices in $X$ to which the element can be mapped. Since the total number of elements in $G\times G$ is 4, the total number of possible mappings is $2\times 2\times 2\times 2 = 2^{4} = 16.$

In general, if $A$ and $B$ are finite sets, then there are $|B|^{|A|}$ different mappings/functions from $A$ to $B$. Does this help answer your question?
 
  • #3
ok, I think I understand now, thank you.
 

Related to Exploring Finite Group Theory: Finding the Upper Bound of Groups of Order

1. What is finite group theory?

Finite group theory is a branch of mathematics that studies the properties and structures of groups with a finite number of elements. It is used to understand and classify groups of objects that have a finite number of symmetries or transformations.

2. What is the upper bound of a group of order?

The upper bound of a group of order refers to the maximum number of elements that a group can have. It is determined by Lagrange's theorem, which states that the order of a subgroup must divide the order of the group. Therefore, the upper bound of a group of order is the highest number that is a divisor of the order of the group.

3. How is the upper bound of a group of order calculated?

The upper bound of a group of order can be calculated using various methods, such as the Sylow theorems, Cauchy's theorem, and the classification of finite simple groups. These methods involve analyzing the prime factorization of the order of the group and applying specific criteria to determine the upper bound.

4. Why is finding the upper bound of groups of order important?

Finding the upper bound of groups of order is important because it allows us to understand the structure and properties of finite groups. It also helps in classifying and organizing groups into different categories based on their order and other characteristics. Additionally, the upper bound can provide insights into other areas of mathematics, such as group theory and abstract algebra.

5. What are some real-world applications of finite group theory?

Finite group theory has various applications in different fields, such as cryptography, chemistry, and physics. In cryptography, it is used to develop secure encryption algorithms. In chemistry, it is used to study the symmetries of molecules. In physics, it is used to describe the symmetries of particles and their interactions. It also has applications in coding theory, combinatorics, and graph theory.

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