Infinite groups with elements of finite order

"Infinite", as in having an infinite number of elements? The circle group U(1) comes to mind, having all of the cyclic groups Z_n as subgroups.

 

How about (0,1) , in Z(+)Z/2 ?

 

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.

 

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.

 

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.

 

Sounds like you're missing some basic definitions:

http://en.wikipedia.org/wiki/Direct_product_of_groups

 

Eddyski3:

The group Z(+)Z/2 is the direct sum of groups, which carries the operation of the first group in the first component, and the operation of the second group in the second component, so that:

(a,b)+(a',b'):= (a+'_Za', b+'_Z/2b') , so you add the first

components as if you were in Z, and the other components as if you were in Z/2,

so (0,1)+(0,1)=(0+0,1+1)=(0,0).

Ben:
You're right, I missed your point; any rational multiple of pi will have finite order.