Finite group of even order has elements of order 2

By assuming otherwise, we can reach a contradiction and prove that an element of order 2 must exist in G.
  • #1
1MileCrash
1,342
41
[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.
Contradiction
5.) Therefore, there exists a y in G/{1} such that y = y^-1, and thus has order 2.
 
Physics news on Phys.org
  • #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.
Contradiction
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.
 

Related to Finite group of even order has elements 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.

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
569
  • Calculus and Beyond Homework Help
Replies
1
Views
600
  • Calculus and Beyond Homework Help
Replies
6
Views
909
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
793
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
934
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
Back
Top