Doesn't it differ by the fact that there exists no integer ##n## for which ##a^n=e##, where ##a## is an element of the group besides e, the neutral element?fresh_42 said:What is a infinite cyclic group and how does it differ from a finite one?
Yes. So we have ##C = \langle a^n\,\vert \,n \in \mathbb{Z} \rangle = \{\ldots , a^{-2},a^{-1},e,a,a^2,\ldots\}##.Mr Davis 97 said:Doesn't it differ by the fact that there exists no integer ##n## for which ##a^n=e##, where ##a## is an element of the group besides e, the neutral element?
An infinite cyclic group is a mathematical structure that consists of a set of elements and a binary operation that follows the properties of closure, associativity, identity, and invertibility. It is generated by a single element, and the group can be infinitely generated by repeatedly applying the binary operation to the generator.
An infinite group means that it contains an infinite number of elements, while a cyclic group means that it is generated by a single element and all other elements of the group can be expressed as powers of that element.
If an infinite cyclic group does not have any non-trivial proper subgroups, it means that the group is simple, which can make it difficult to study or apply in certain mathematical contexts. Non-trivial proper subgroups provide a way to break down the group into smaller, more manageable parts.
One example of an infinite cyclic group with non-trivial proper subgroups is the group of integers under addition, denoted as ℤ. The subgroups of this group include the set of even integers, the set of multiples of 3, and the set of multiples of any integer.
The fact that every infinite cyclic group has non-trivial proper subgroups is related to the concept of group factorization, where a group can be broken down into smaller subgroups. It also relates to the study of group theory and its applications in various fields such as cryptography, physics, and computer science.