# Finite group of even order has elements of order 2

• 1MileCrash
By assuming otherwise, we can reach a contradiction and prove that an element of order 2 must exist in G.
1MileCrash
[The homework format does not appear on mobile]

Problem: Show that a finite group of even order has elements of order 2

Attempt:
The book gives a suggested approach that lead me to write the most round about, ugly proof I've ever written.

Can't I just say:
1.) If G has even order, G/{1} has odd cardinality.
2.) Assume that no elements of G/{1} has order 2.
3.) Then for each x in G/{1}, x^-1 is a distinct element of G/{1}.
4.) Then G/{1} has even cardinality.
5.) Therefore, there exists a y in G/{1} such that y = y^-1, and thus has order 2.

1MileCrash said:
[The homework format does not appear on mobile]

Problem: Show that a finite group of even order has elements of order 2

Attempt:
The book gives a suggested approach that lead me to write the most round about, ugly proof I've ever written.

Can't I just say:
1.) If G has even order, G/{1} has odd cardinality.
2.) Assume that no elements of G/{1} has order 2.
3.) Then for each x in G/{1}, x^-1 is a distinct element of G/{1}.
4.) Then G/{1} has even cardinality.
5.) Therefore, there exists a y in G/{1} such that y = y^-1, and thus has order 2.

That is the standard method of showing that a finite group of even order has an element of order 2.

## 1. What is a finite group?

A finite group is a mathematical structure consisting of a set of elements and an operation that combines any two elements to form a third element. The elements in a finite group are limited in number and the operation follows certain rules, such as closure, associativity, and identity.

## 2. What does it mean for a group to have even order?

The order of a group refers to the number of elements in the group. If a group has an even order, it means that the number of elements in the group is divisible by 2.

## 3. What does it mean for an element to have order 2?

The order of an element in a group is the smallest positive integer n such that when the element is multiplied by itself n times, the result is the identity element of the group. Therefore, an element of order 2 means that when it is multiplied by itself twice, the result is the identity element.

## 4. Why does a finite group of even order have elements of order 2?

This is a fundamental property of finite groups of even order. It is a consequence of the Lagrange's theorem, which states that the order of any subgroup of a finite group must divide the order of the group. Since the order of a group is even, any subgroup of this group must have an order that is a divisor of the even number. This means that there must be elements of order 2 in the group.

## 5. Can a finite group of even order have elements of order other than 2?

Yes, a finite group of even order can have elements of order other than 2. However, the number of elements of order 2 in such a group must be even. This is because the identity element always has order 1, leaving an even number of elements of the group to have order other than 1.

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