# Two conjugate elements of a group have the same order PROOF

• RJLiberator
In summary: Therefore, x and y are conjugate elements and share the same order.In summary, conjugate elements x and y in a group G have the same order if and only if x^n = e and y^n = e for some n ∈ N. The definition of conjugate elements is that there exists a group element g such that gaga^-1 = b. By this definition, it can be proven that x and y are conjugate elements and thus have the same order if x^n = e and y^n = e.

Gold Member

## Homework Statement

Let x and y be conjugate elements of a Group G. Prove that x^n = e if and only if y^n = e, hence x and y have the same order.

## Homework Equations

Conjugate elements : http://mathworld.wolfram.com/ConjugateElement.html

## The Attempt at a Solution

Since y is a conjugate of x, there exists z ∈G such that y = (zxz^-1).

If x^n = e, then y^n = (zxz^-1)^n = (zx^nz^-1) = (zez^-1) = zz^-1 = e.

Similiarly, if y^n = e, then e=y^n = (zxz^-1)^n = (zx^nz^-1). Multiplying on the left by z^-1 and on the right by z we see
z^-1ez = z^-1zx^nz^-1z and so e = ex^ne = x^n

Done.My concern is in the definition of conjugate elements. If x and y are conjugate elements of a group G, does that necessarily mean y is a conjugate of x?

I've supplied the definition of conjugate elements in the 'relevant equations' part of this thread.

I don't like that definition because it never clearly states the necessary and sufficient conditions for being conjugate. It also blurs the distinction between the - at first - very different statements 'a and b are conjugate' and 'a is conjugate to b'. Use this definition instead:

If a and b are elements of group G then we say that a is conjugate to b iff there exists ##g\in G## such that ##gag^{-1}=b##.

It is then simple to prove that a is conjugate to b iff b is conjugate to a.

Hence, we can refer to two elements as being conjugate, without specifying an order, ie the statement 'a and b are conjugate in G' is well-defined, and means 'a is conjugate to b', which is the same as 'b is conjugate to a'.

RJLiberator
So, when the question states "conjugate elements" we can think of the definition as

andrewkirk said:

If a and b are elements of group G then we say that a is conjugate to b iff there exists ##g\in G## such that ##gag^{-1}=b##.

The way the question is worded, and by this definition, then I can safely assume that there exists z ∈G such that y = (zxz^-1) and my proof follows.

## What is the definition of conjugate elements in a group?

Conjugate elements in a group are elements that are related to each other through a similarity transformation. This means that they have the same underlying structure, but may differ in specific properties such as order or exponent.

## Why is it important to prove that two conjugate elements have the same order in a group?

Proving that two conjugate elements have the same order in a group is important because it allows us to understand the structure of the group and its subgroups. It also helps us to identify important properties and relationships between elements within the group.

## What is the process for proving that two conjugate elements have the same order in a group?

The process for proving that two conjugate elements have the same order in a group involves showing that they have the same structure, and then using this to demonstrate that they have the same order. This often involves using the group's defining properties and the properties of conjugate elements.

## What are some common examples of groups where conjugate elements have the same order?

Some common examples of groups where conjugate elements have the same order include symmetric groups, dihedral groups, and cyclic groups. In these groups, conjugate elements often have the same order due to the specific structure and properties of the group.

## What are the implications of two conjugate elements having the same order in a group?

The implications of two conjugate elements having the same order in a group can vary depending on the specific group and context. However, it often indicates a strong relationship between the elements and can provide insights into the structure and properties of the group.