I've just started the beginnings of abstract algebra and came across the concept of infinite cyclic groups. I've read that such a group can be represented with the group of integers under addition (which makes sense). However, I was wondering if this infinite cyclic group could also be viewed through the lens of the group of real numbers under addition.
If they cannot, could someone please explain why? (My math literacy is not overly sophisticated so I would greatly appreciate simplicity if possible).