Proving Order 2 Element in Finite Even Group G

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In summary, the conversation discusses the proof of the statement that any finite group of even order contains an element of order 2. The individual is frustrated because they know this statement can be proven using Cauchy's Theorem, but they must still prove it using group axioms. They are currently stuck and are looking at the subset of the group containing elements that are not of order 2. They believe this subset must have an even number of elements and that the complement of this subset must include at least the identity of the group. Therefore, there must be at least one more element in the complement, fulfilling the role of the element of order 2. The individual is unsure if their reasoning is correct and seeks confirmation.
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BSMSMSTMSPHD
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I'm working on the proof of the following statement: Prove that any finite group G of even order contains an element of order 2.

What frustrates me about this is that I know from a previous class that this statement immediately follows from Cauchy's Theorem. So, in a way, I don't want to prove it using group axioms.

But I hafta, so whining aside, I will carry on. But I seem to be stuck.

So far, I am looking at the subset of G that contains elements that are not of order 2. That is:

[tex] H = \{ g \in G \ | \ g \neq g^{-1} \} [/tex]

I think that this set must have an even number of elements since for each h in H, the inverse of h must also be in H, and, they can't be the same.

Does this make sense?

Then, H complement must include at least the identity of G. So, since G is the union of H and H complement, it seems that there must be at least one more element, a, in H complement, so that the total number of elements in G is indeed even.

Thus, a fulfills the role of the element of order 2.

Mathematical casualness aside, does this sound good? Thanks...

EDIT: Ok, maybe "painstaking" was a bit overdramatic...
 
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  • #2
That's perfectly fine, why the lack of confidence?
 
  • #3
Oh, it's a way of life.:eek:
 
  • #4
snap out of it. be critical of your work but accept the fact that not being able to find an error sometimes means there are no errors and you are right on the money :bugeye:
 

1. What is the definition of an "order 2 element" in a finite even group?

An order 2 element in a finite even group G is an element that, when multiplied by itself, results in the identity element of the group. In other words, the order of the element is 2.

2. How do you prove the existence of an order 2 element in a finite even group G?

To prove the existence of an order 2 element in a finite even group G, we can use the fact that every even group has at least one element of order 2. This is because the group contains both even and odd numbers, and the product of any two odd numbers is an even number. Therefore, there must be at least one element in the group that, when multiplied by itself, results in the identity element.

3. Can an order 2 element exist in a finite odd group?

No, an order 2 element cannot exist in a finite odd group. This is because in a finite odd group, all elements have an odd order. The order of an element is defined as the smallest positive integer n such that the element raised to the power of n is equal to the identity element. Since the order of an odd group is always odd, there can be no element with an order of 2.

4. Is the order of an element in a finite even group always a multiple of 2?

Yes, the order of an element in a finite even group is always a multiple of 2. This is because in a finite even group, the order of the group itself is always even. Therefore, all elements in the group must have an order that is a factor of the group's order, and since the group's order is even, the order of each element must also be even.

5. How does the existence of an order 2 element in a finite even group impact the group's structure?

The existence of an order 2 element in a finite even group has several implications on the group's structure. Firstly, it means that the group is not cyclic, as a cyclic group can only have elements with orders that are factors of the group's order. Additionally, the presence of an order 2 element means that the group is not abelian, as the multiplication of the group is not commutative. Finally, the existence of an order 2 element also affects the group's center, as the center of a group always contains elements of order 2.

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