Group Theory: Finite Group Has Prime Order Element

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In summary: So you can construct elements with any order you want.In summary, to show that a finite group contains at least one element with prime order, we can use the fact that a non-identity element with composite order can be broken down into smaller orders, including a prime order. This allows us to construct an element with the desired prime order.
  • #1
alexmahone
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Show that if G is a finite group, then it contains at least one element g with |g| a prime number. (|g| is the order of g.)

Hints only as this is an assignment problem.
 
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  • #2
Let $g$ be an arbitrary non-identity element of the group, with order $n > 1$. Suppose $n$ is not prime, and so $g^n = 1$ but $g^k \ne 1$ for all $1 \leq k < n$. If $n$ is not prime, it is divided by some prime $p$. Can you find an element of the group which has order $p$?
 
  • #3
It's not *quite* true- if $G$ has order 1...
 
  • #4
Bacterius said:
Let $g$ be an arbitrary non-identity element of the group, with order $n > 1$. Suppose $n$ is not prime, and so $g^n = 1$ but $g^k \ne 1$ for all $1 \leq k < n$. If $n$ is not prime, it is divided by some prime $p$. Can you find an element of the group which has order $p$?

Let $n=ap$.

$g^n=g^{ap}=(g^a)^p$

So, $(g^a)^p=1$ ------------- (1)

Let us consider $(g^a)^m$ where $1\le m<p$.

Multiplying this inequality by $a$, we get $a\le am<ap$ or $a\le am<n$.

We know that $g^k\ne 1$ for all $1\le k<n$. Since $am<n$, $g^{am}\ne 1$.

So, $(g^a)^m\ne 1$ where $1\le m<p$ ------------- (2)

From (1) and (2), the order of $g^a$ is $p$.

Is this correct?
 
  • #5
Alexmahone said:
Let $n=ap$.

$g^n=g^{ap}=(g^a)^p$

So, $(g^a)^p=1$ ------------- (1)

Let us consider $(g^a)^m$ where $1\le m<p$.

Multiplying this inequality by $a$, we get $a\le am<ap$ or $a\le am<n$.

We know that $g^k\ne 1$ for all $1\le k<n$. Since $am<n$, $g^{am}\ne 1$.

So, $(g^a)^m\ne 1$ where $1\le m<p$ ------------- (2)

From (1) and (2), the order of $g^a$ is $p$.

Is this correct?

That's right :) using this technique you can easily construct a group element that has the desired prime order, as you have shown. In general, if $g$ has order $nm$ then $g^n$ has order $m$ and symmetrically $g^m$ has order $n$.
 

FAQ: Group Theory: Finite Group Has Prime Order Element

What is a finite group?

A finite group is a mathematical structure that consists of a set of elements and a binary operation that combines any two elements to form a third element. The set of elements is finite, meaning it has a limited and finite number of elements.

What does it mean for a finite group to have a prime order element?

A prime order element in a finite group is an element whose order (the smallest positive integer n such that g^n = e, where e is the identity element) is a prime number. This means that the element can only be multiplied by itself a certain number of times before it results in the identity element.

How can you determine if a finite group has a prime order element?

To determine if a finite group has a prime order element, you can use the order of the group and the properties of prime numbers. If the order of the group is a prime number, then all elements in the group will have a prime order. If the order of the group is not a prime number, you can use the Lagrange's theorem to find the possible orders of the elements in the group. If any of these orders is a prime number, then the group has a prime order element.

What is the significance of a finite group having a prime order element?

A finite group having a prime order element has several implications. For example, it can simplify certain calculations and proofs involving the group, as the order of the element is a prime number. It also allows for a deeper understanding of the structure of the group and its subgroups. Additionally, prime order elements have important applications in cryptography and coding theory.

Can a finite group have more than one prime order element?

Yes, a finite group can have more than one prime order element. In fact, if a group has a prime order, then all of its elements will have prime orders. However, a group can also have multiple elements with different prime orders. For example, the group Z/12Z has two prime order elements, 5 and 7, as both are relatively prime to 12.

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