In my notes for class it says that elements of the additive group Q have either order 1 or infinity. The former part of this is obviously true (since id element will have order of 1). But why do all other elements have to have order infinity? If an element of the group Q is not of order infinity, does that mean it can cover the entire group?
I know that elements in infinite cyclic groups have order of either 1 or infinity, but I'm not sure how to prove that for the additive group Q since it is not cyclic.
Thanks.
Best Answer
Hint: What is the additive identity in $\mathbb{Q}$? Now, if an element has finite order, that means that the sum of it with itself some finite number of times is equal to that additive identity. Is that possible?