Number of elements of order $2$ in Abelian groups of order $2^{n}$

I'm self studying some group theory and one of the exercises I came across:

Question: Prove that an Abelian group of order $2^{n}, n \in \mathbb{N}$ must have an odd number of elements of order 2.

I'm not sure how to approach this problem. Any hints would be appreciated. Prefer no complete answers but could use one for self checking. Thank you.

• Hint: You don't need the group to be abelian, and you just need the order to be even. Try to pair up elements with their inverses. – Tobias Kildetoft Dec 26 '15 at 9:10
• Interesting theorem in this context: if $G$ is a finite group and its order is a multiple of prime $p$ then also $a+1$, where $a$ stands for the number of elements having order $p$, is a multiple of $p$. So here it is actually enough if $G$ is a finite group (as @TobiasKildetoft remarks not necessarily abelian) and its order is even. – drhab Dec 26 '15 at 10:05

Hint: The product of any two order-two elements is another element of order two (or the identity).

A much more direct hint:

The set of elements of order two, joined with the identity element, form a subgroup.

The anwser:

The order of that group must divide $2^n$ and so must be $2^k$ for some $k<n$. Take out the identity and you have that the set of order-two elements has $2^k-1$ elements.

• Fixed! Careless of me. – Trold Dec 26 '15 at 9:00

Hint: The elements of order exactly four come in pairs, namely $g$ and $g^3$. The elements of order $8$ come in quartets, $g, g^3, g^5, g^7$. The elements of order $16$ come in octets.

The element of order $1$ is lonely.

• This shows there's an even number of elements of order $2^n$ for $n\geq2$. But how does that show there's an odd number of elements of order $2$? – Gregory Grant Dec 26 '15 at 8:09
• @GregoryGrant The number of elements in the group is even, and the set of elements of order $> 2$ has even size. So the set of elements of order 1 and 2 has even size. – user296602 Dec 26 '15 at 8:12

Hint: define a relation on $G$ by $$a\sim b\quad\text{if and only if}\quad b=a\text{ or }b=a^{-1}$$ Prove this is an equivalence relation and that the equivalence classes have one or two elements. Which ones have just one element?