Prove of disprove: there exists a finite group with an element of infinite order.

I am not convinced this holds, because a finite group will have the form

, where

is the identity element. Since groups are closed under their operation, I would think each element would eventually have to be the identity for some

; namely,

for some

.

Not sure though, we just introduced the concept of order yesterday. I looked ahead in the book and read a little about cyclic groups, but I don't think any knoweldge of that is needed for this.