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Main Question or Discussion Point
Can anyone think of an example of an infinite group that has elements with a finite order?
But in my example they are the same. For example, [itex]e^{i \pi /4}[/itex] has finite order in U(1), while [itex]e^{i \pi / \sqrt{2}}[/itex] has infinite order.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.
Yet another construction is to take any finite group G and construct the group [itex]H = G \times \mathbb{Z}[/itex]. Then H has elements of finite order given by (g, 0), and all elements of the form (g, z) for [itex]z \neq 0[/itex] have infinite order.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?