# Are Even Permutations a Subgroup of D4?

• POtment
In summary, the conversation discusses identifying and proving the even permutations in a subgroup of D4, which involves understanding the definition of an even permutation and using it to show that the set of even elements forms a subgroup under D4. The conversation also includes a discussion on the properties of groups and how to show that the set of even permutations is a subgroup.
POtment

## Homework Statement

Consider the group D4 (rigid motions of a square) as a subgroup of S4 by using
permutations of vertices. Identify all the even permutations and show that they form a subgroup of D4.

## The Attempt at a Solution

I think I have the permutations of correct. They are: (1,2,3,4), (1,3)(2,4), (1,4,3,2), (1), (2,4), (1,3), (1,2)(3,4), (1,4)(2,3). If this is correct, then the only one that is not even is (1).

Can someone check my work thus far? I know how to go about proving it is a subgroup assuming the rest is correct.

Your list of group elements looks correct. The conclusion that (1) is not even is dead wrong. Better reread the definition of an 'even' permutation. There are four even permutations in there and four odd ones.

The four even then would be (13)(24), (12)(34), (14)(23), and (1), correct?

Is 1 even because it is 0 transpositions?

POtment said:
The four even then would be (13)(24), (12)(34), (14)(23), and (1), correct?

Is 1 even because it is 0 transpositions?

Yes.

POtment said:
The four even then would be (13)(24), (12)(34), (14)(23), and (1), correct?

Is 1 even because it is 0 transpositions?

Yes.

I'm having some problems completing the proof that the set of even elements forms a group under D4. I do know that if G is any group of permutations then the set of all even permutations G form a subgroup of G, but I'm not sure how to prove that. Does that seem the like the easiest way to go about it?

If a*b is the product of the two permutations a and b, what can you say about whether a*b is even or odd in terms of the even or oddness of a and b? Then to show the evens are a subgroup, show it's closed, has inverses, has an identity etc.

OK, that makes sense. I've completed this problem. Thanks!

## 1. What is a permutation?

A permutation is a mathematical concept that refers to the rearrangement of a set of objects or elements in a specific order. It is often represented as a function that maps elements from one set to another, while maintaining the same number of elements in both sets.

## 2. What is a subgroup?

A subgroup is a subset of a larger mathematical group that retains the same structure and properties as the original group. It is formed by a smaller set of elements that still satisfy the group's defining rules and operations.

## 3. How do permutations form a subgroup?

Permutations form a subgroup because they satisfy the properties of closure, associativity, and identity within a larger group. This means that the composition of two permutations will always result in another permutation, and the identity permutation (which does not change the order of elements) is also included in the subgroup.

## 4. Why is the subgroup formed by permutations important?

The subgroup formed by permutations is important in many areas of mathematics, including group theory, combinatorics, and abstract algebra. It allows for easier analysis and manipulation of permutations, as well as providing a useful tool for solving various problems and equations.

## 5. Can any set of permutations form a subgroup?

No, not all sets of permutations will form a subgroup. For a set of permutations to form a subgroup, it must satisfy the properties of closure, associativity, and identity. Additionally, the set must also contain the inverse of each permutation, meaning that there is a permutation that can "undo" the original permutation. If these conditions are not met, the set will not form a subgroup.

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