Question

Prove that any Abelian group of order 2^n (n greater than or equal to 1) must have an odd number of elements of order 2.

I know it has something to do with the Fundamental Theorem of Finite Abelian groups, or at least i think it does.

Answer 1

Let H be the set of all elements from your abelian group that has order at most 2

show that H is a subgroup of your abelian group.

if it is a subgroup then is has to have even order (Lagrange's theorem). Remove e (the identity) from H then you have the collection of elements or order 2 (and there is an odd number of them)