- #1
jessicaw
- 56
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a group or a cyclic group of finite order can i just repeatedly write down the repeated elements and form a very large even infinite group?
HallsofIvy said:What do you mean by "write down repeating elements". If the group is of finite order, then every member of it has finite order. Eventually, "repeating elements" get you back to the identity and then you get nothing new.
A group of finite order can be infinitely large because the order of a group only reflects the number of elements in the group, not the size or magnitude of the elements themselves. In other words, a group's order does not determine its size, but rather the number of distinct elements it contains.
Yes, the group of real numbers under addition is an infinite group of finite order. Although the set of real numbers is infinitely large, the group only has two distinct elements: 0 and all other real numbers. Therefore, the order of this group is 2, but it is infinitely large.
A group's order is determined by its operation and not the elements themselves. Therefore, even if a group has a finite number of elements, the operation can still generate an infinite number of combinations, resulting in an infinite group of finite order.
The fact that a group can have a finite order and be infinitely large highlights the importance of understanding the structure and properties of a group beyond just its order. This concept also challenges traditional notions of size and infinity, showing that they can coexist in mathematical structures.
No, the concept of a group with a finite order and infinite size is a purely mathematical concept and does not have a physical equivalent. In the physical world, objects and quantities are limited and cannot be infinitely large, making the idea of a finite group with an infinite order impossible to manifest.