# Group Theory and Order of Elements

• July 2nd 2011, 11:26 AM
Kopeck
Group Theory and Order of Elements
Hello. I have a question about the order of elements in a group:

let G be a group of order 2n. Show that the number of elements in G with order 2 is odd.

Here is what I have done so far:

Let H be the subset of G consisting of the identity and all the elements of G with order 2. Then H is a subgroup of G and so, by Lagrange's Theorem, the order of H divides the order of G.

My strategy was to use this to arrive at the order of H is even, completing the proof. But I am stuck at this point.
• July 2nd 2011, 11:41 AM
Isomorphism
Re: Group Theory and Order of Elements
Hint: Prove that $|\{ a \in G | a \neq a^{-1} \}|$ is even. Check the parity of what remains, and notice that $e \notin \{ a \in G | a \neq a^{-1} \}$
• July 2nd 2011, 12:21 PM
Kopeck
Re: Group Theory and Order of Elements
Oh my goodness. This makes it very clear. Thank you for the hint Isomorphism.