Does Group Cardinality Determine Element Order?

In summary, the conversation discusses the concept of order in a group and the relationship between the order of the group and the order of an element. It is stated that the order of a group is simply the number of elements in the group, while the order of an element is the smallest possible integer n such that the element raised to the nth power equals the identity element. The conversation also questions whether this relationship can be used to show that g^n = e for a group G with n elements.
  • #1
RJLiberator
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Homework Statement


If G is a group with n elements and g ∈ G, show that g^n = e, where e is the identity element.

Homework Equations

The Attempt at a Solution



I feel like there is missing information, but that cannot be.
This seems too simple:
The order of G is the smallest possible integer n such that g^n = e. If no such n exists, then G is of infinite order.

From this definition of order can we simply state that since G is a group with 'n' elements then there must exist an n such that g^n = e ?

order is denoted as °(g)
So
°(g) = n ==> g^n = e.
 
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  • #2
RJLiberator said:

Homework Statement


If G is a group with n elements and g ∈ G, show that g^n = e, where e is the identity element.

Homework Equations

The Attempt at a Solution



I feel like there is missing information, but that cannot be.
This seems too simple:
The order of G is the smallest possible integer n such that g^n = e. If no such n exists, then G is of infinite order.
Aren't you mixing up G and g here? You're describing the definition of the order of the element g, not the order of the group G.

From this definition of order can we simply state that since G is a group with 'n' elements then there must exist an n such that g^n = e ?

order is denoted as °(g)
So
°(g) = n ==> g^n = e.
 
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  • #3
Where did you get this: "The order of G is the smallest possible integer n such that g^n = e"? The definition of "order of a group" is simply the number of elements in the group (the cardinality of the underlying set). It looks to me like you are being asked to prove the statement you give.
 

FAQ: Does Group Cardinality Determine Element Order?

1. What is Group Theory?

Group Theory is a branch of abstract algebra that deals with the properties and structure of mathematical groups. A group is a set of elements that follow a specific set of rules, such as closure, associativity, and the existence of an identity element and inverses.

2. What does the notation g^n = e mean?

The notation g^n = e represents the group operation of raising an element g to the power of n, resulting in the identity element e. This means that when an element is repeatedly combined with itself n times, it will eventually result in the identity element.

3. What is the significance of proving g^n = e in Group Theory?

Proving g^n = e is significant because it demonstrates the ability to manipulate elements within a group and shows that every element in the group has an inverse. It also helps to establish the structure and properties of the group being studied.

4. How is the proof of g^n = e typically approached?

The proof of g^n = e is typically approached using mathematical induction, where the statement is first proven for n = 1, and then it is shown that if the statement holds for n = k, then it also holds for n = k+1. This establishes the proof for all positive integer values of n.

5. Are there any real-world applications of Group Theory and the proof of g^n = e?

Yes, Group Theory has many applications in fields such as physics, chemistry, and computer science. For example, it is used in quantum mechanics to study the properties of particles and in cryptography to ensure the security of data encryption. The proof of g^n = e is also used in various algorithms and protocols in computer science.

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