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A Group of Even Order

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Eli
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#1

A group is an algebraic structure that is closed under binary operation, associative, each of its element has an inverse and there is a common identity element.

Problem

Let be a finite group of even order. Show that has an element such tha

Proof

Let be the group of even order.

Suppose that each element of is paired with its inverse ( Note that the identity element is paired to itself)

Then consists of the union of all of the pairs

Now,

Suppose that no element other than identity, is paired to itself

Let the number of pairs excluding the identity be , then (since there are pairs + the identity)

But, this makes the order of odd, a contradiction, and thus there must be at least one element that is paired with its inverse (this makes the order of even)

Let be paired with its inverse, then , and hence
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