Please help - in my notes, it is the group $V$ itself. I just want to confirm this. Can you also explain and give an example if that is possible?
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Every abelian group is its own center. If you look at the definitions you will see this.
You know, this group has only 4 elements. You could just multiply them out. Really, there are only three worthwhile elements, as one is an identity and that commutes with everything.
It would take less than 5 minutes, and you could do it Cayley style or full-on multiplication table style. It would probably even be good for you, as you might get a feel for what groups really are.