The discussion revolves around the properties of infinite cyclic groups, specifically focusing on the existence of non-trivial proper subgroups. Participants are exploring the differences between finite and infinite cyclic groups and the implications of these differences on subgroup structure.
Some participants have provided insights into the nature of infinite cyclic groups and have begun to explore what constitutes a non-trivial subgroup. There is an ongoing inquiry into the characteristics of these groups and the conditions that define their subgroups.
Participants are questioning the assumptions related to subgroup definitions and the implications of group order, particularly in the context of infinite groups. There is an emphasis on understanding the structure of infinite cyclic groups without reaching a definitive conclusion.
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?