# Proving the Subgroup Property of Even Permutations in Permutation Groups

• PhysicsUnderg
In summary, the conversation discusses the proposition that states that if G is a group of permutations, then the set of all even permutations in G forms a subgroup of G. The conversation then goes on to discuss the proof of this statement, with the hint that if H is a subgroup of G, then for any a and b in H, the product ab must also be in H. It is concluded that this is true for even permutations, as the product of two even permutations is also an even permutation. The conversation also mentions that for H to be a subgroup, it must contain the identity element and its inverse, and poses questions about the identity and inverse of even permutations.
PhysicsUnderg

## Homework Statement

Show that if G is any group of permutations, then the set of all even permutations in G forms a subgroup of G.

I am not sure where to start - I know there is a proposition that states this to be true, but I know that is not enough to prove this statement.

Hi PhysicsUnderg!

Hint: if H is a subgroup of G, then for any a and b in H, the product ab must also be in H.

Is it really that simple? lol This is what I was thinking, but I wasn't sure how to connect the idea to permutations. Can I just say "a is an even permutation" and "b is an even permutation" thus "a*b is also an even permutation"? Because, if this is true and if I assume that a and b are elements of H, then ab is an element of H and is an even permutation, so G has a subgroup of even permutations. Also, for H to be a subgroup, the identity element must be contained in H, as well as an inverse. How do you connect this to permutations?

Hi PhysicsUnderg!
PhysicsUnderg said:
Is it really that simple?

Yes!

Sometimes, maths really is that simple!
… Also, for H to be a subgroup, the identity element must be contained in H, as well as an inverse. How do you connect this to permutations?

You ask "Is the identity an even permutation? What is the inverse of an even permutation?"

## 1. What is the subgroup property of even permutations in permutation groups?

The subgroup property of even permutations in permutation groups states that the set of even permutations in a given permutation group forms a subgroup. This means that the product of any two even permutations in the group will also be an even permutation.

## 2. How do you prove the subgroup property of even permutations?

The subgroup property can be proved using the one-step subgroup test, which states that in order for a subset of a group to be a subgroup, it must contain the identity element and be closed under the group's operation. In the case of even permutations, this means that the product of any two even permutations must also be an even permutation.

## 3. What is the significance of the subgroup property of even permutations in permutation groups?

The subgroup property of even permutations is significant because it allows us to easily identify a subgroup within a larger permutation group. This can be useful in solving problems involving permutations, as it allows us to focus on a smaller subset of the group.

## 4. Can the subgroup property of even permutations be extended to odd permutations?

No, the subgroup property only applies to even permutations. This is because the product of an even and an odd permutation always results in an odd permutation, breaking the closure property required for a subgroup.

## 5. Are there any other properties of even permutations in permutation groups?

Yes, even permutations also have the property of being able to be expressed as a product of transpositions. This means that any even permutation can be written as a sequence of swapping two elements at a time. Additionally, the set of even permutations forms an abelian subgroup within the larger permutation group, meaning that the order in which the permutations are applied does not affect the end result.

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