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eddyski3
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Can anyone think of an example of an infinite group that has elements with a finite order?
Bacle said:Ben:
I imagine eddy wants the elements to have finite order under the operation of the ambient group, not under the operation of the subgroups.
eddyski3 said:Yes, as in having an infinite number of elements. What if we wanted a group with order infinity that had a relatively small number of elements of finite order?
An infinite group with elements of finite order is a mathematical structure that consists of an infinite number of elements, each of which has a finite order. This means that when an element is multiplied by itself a certain number of times, it will eventually result in the identity element of the group.
Infinite groups with elements of finite order have important applications in various areas of mathematics and physics, including group theory, number theory, and quantum mechanics. They also have connections to the study of symmetry and patterns.
No, not all elements in an infinite group have finite order. There can be elements in an infinite group that have infinite order, meaning that when multiplied by themselves any number of times, they will not result in the identity element of the group.
The order of an element in an infinite group can be determined by finding 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. This is known as the order of the element.
Yes, there are many real-life examples of infinite groups with elements of finite order. One common example is the group of rotations in three-dimensional space, where the elements represent rotations by a certain angle around a fixed axis. These rotations have finite order, as they can only be repeated a certain number of times before returning to the starting position.