How do I proof that groups of an even order must have an element of order 2? I have a vague idea, but I don't know how to put my idea together. Aside from identity, there are an odd number of elements in my group. So one element will not have a partner and will have to be multiplied by itself to cancel out. That element must have an order of 2 such that its square = identity. But how can I create the scenario where all elements have to pair up and cancel out? Thanks in advance.