Prove that for every odd integer n, there is a group... Prove that for every odd integer n, there is a group with exactly n elements of order 2.
I am new to abstract algebra and proofs in general. Proving existence is difficult for me, but I will show what I have.
$$n = 2k + 1$$ $$k \in \mathbb{Z^+}$$
$$\exists G \supset H = (a_1 , a_2 , ..., a_n, e)$$
$$s.t. |a_1| = |a_2| = ... = |a_n| = 2$$
My instinct is to use Lagrange's Theorem, although I would like to try another method to prove this as we have not gone over cosets in class yet. If someone could just give me a hint of how to start, it would be very beneficial. Thank you.

Hint: look at the dihedral group of 2n elements, where n is odd.
