# On the edge of completing this proof about the order of an element

• jdinatale
In summary, the order of an element in a group is the smallest positive integer n such that when the element is multiplied by itself n times, it results in the identity element of the group. To determine the order of an element in a group, you can repeatedly multiply the element by itself until you reach the identity element. The number of times you need to multiply the element is the order of that element. The order of an element is important in group theory because it helps us understand the structure and properties of a group, classify groups, and identify important subgroups. Two elements in a group can have the same order, and in a cyclic group, all elements have the same order as the order of the group itself. The order of an

## The Attempt at a Solution

Based on my attempt above, I have demonstrated that the order is less than or equal to n. I am certain that it is actually n, but I'm not sure which theorem I can grab finish this thing off. Any tips for pointing me in the right direction?

(bab^(-1))^m is equal to b*a^m*b^(-1) for any m. You've shown that. If that's equal to e for m<n, then was the order of a really n?

## What is the order of an element in a group?

The order of an element in a group is the smallest positive integer n such that when the element is multiplied by itself n times, it results in the identity element of the group.

## How do you determine the order of an element in a group?

To determine the order of an element in a group, you can repeatedly multiply the element by itself until you reach the identity element. The number of times you need to multiply the element is the order of that element.

## Why is the order of an element important in group theory?

The order of an element is important in group theory because it helps us understand the structure and properties of a group. It also helps us classify groups and identify important subgroups.

## Can two elements in a group have the same order?

Yes, it is possible for two elements in a group to have the same order. For example, in a cyclic group, all elements have the same order as the order of the group itself.

## How does the order of an element relate to the order of a group?

The order of an element can be a factor of the order of a group. For example, in a cyclic group, the order of an element must divide the order of the group. Additionally, the order of an element can help determine the order of a group in some cases.