Abstract Algebra based on Coset?

Question

Prove that a group of order 12 must have an element of order 2

Answer 1

Here's a general proof that any group of even order must have an element of order 2. .

Consider any element g not the identity. Then either g^(-1) is not equal to g, or else g has order 2. So if there are no elements of order 2, there's an even number of non-identity elements (because you can pair them up), and hence an odd number of elements overall.

QED!