# The permutation induces on the set

• MHB
• mathmari
In summary, the exercise involves making a sketch of a regular tetrahedron and labeling its corners with numbers. Then, for each of the given permutations, we check if it induces a symmetry on the set of vertices and describe the corresponding geometric transformation. The permutations (1 2), (1 2 3), (1 2)(3 4), and (1 2 3 4) induce a reflection, rotation, rotation, and no symmetry respectively. The total number of symmetries of the tetrahedron is 15, consisting of 1 identity, 8 rotations when one vertex is fixed, 3 rotations when two pairs of vertices are switched, and 6 reflections about the midpoint of an
mathmari
Gold Member
MHB
Hey!

I am looking at the following exercise:

Make a sketch of a regular tetrahedron and label the corners with the numbers $1, 2, 3, 4$. For $1\leq i\leq 5$ the permutations $\pi_i \in \text{Sym} (4)$ are defined as follows:
\begin{align*}&\pi_1:=\text{id} \\ &\pi_2:=(1 \ \ 2) \\ &\pi_3:=(1 \ \ 2 \ \ 3) \\ &\pi_4:=(1 \ \ 2)\ (3 \ \ 4) \\ &\pi_5:=(1 \ \ 2 \ \ 3 \ \ 4)\end{align*}
1. For $1 \leq i \leq 5$, check whether there is a symmetry of the tetrahedron that the permutation $\pi_i$ induces on the set of vertices $\{1, 2, 3, 4\}$ and, if so, give a geometric description of this symmetry.
2. Is there for every $\pi \in \text{Sym} (4)$ a symmetry of the tetrahedron, that induces the permutation $\pi$ on the set of vertices $\{1, 2, 3, 4\}$ ?
Could you explain to me the exercise and what I am supposed to do?

I got stuck what it means "induce on a set".

(Wondering)

Hey mathmari!

Those permutations are of themselves merely reorderings of numbers.
At the next step we consider that those numbers represent the corners of the tetrahedron.
The question is then if for instance swapping corners 1 and 2 is actually a symmetry of the tetrahedron.
As it is, it corresponds to a reflection in the plane through corners 3 and 4 that transforms the tetrahedron to itself with corners 1 and 2 swapped.
So they say that the permutation (1 2) "induces" a reflection on the tetrahedron, which is a symmetry of the tetrahedron. (Nerd)

How about the others? What kind of geometric transformation do they represent if any?
Are they symmetries of the tetrahedron? (Thinking)

Klaas van Aarsen said:
Those permutations are of themselves merely reorderings of numbers.
At the next step we consider that those numbers represent the corners of the tetrahedron.
The question is then if for instance swapping corners 1 and 2 is actually a symmetry of the tetrahedron.
As it is, it corresponds to a reflection in the plane through corners 3 and 4 that transforms the tetrahedron to itself with corners 1 and 2 swapped.
So they say that the permutation (1 2) "induces" a reflection on the tetrahedron, which is a symmetry of the tetrahedron. (Nerd)

How about the others? What kind of geometric transformation do they represent if any?
Are they symmetries of the tetrahedron? (Thinking)

As for $\pi_3$, the question is then if corner 1 is changed to corner 2 and corner 2 is changed to corner 3 and corner 3 changed to corner 1 is a symmetry of the tetrahedron.
As it is, it corresponds to a rotation around the axis of corner 4 and the middle of its opposite side.
So they say that the permutation (1 2 3) "induces" a rotation of the tetrahedron, which is a symmetry of the tetrahedron.

As for $\pi_4$, the question is then if swapping corners 1 and 2 and swapping corners 3 and 4 is a symmetry of the tetrahedron.
As it is, it corresponds to a rotation around the axis that passes through the midpoints of two opposite edges.
So they say that the permutation (1 2)(3 4) "induces" a rotation of the tetrahedron, which is a symmetry of the tetrahedron.

As for $\pi_5$, the question is then if corner 1 is changed to corner 2 and corner 2 is changed to corner 3 and corner 3 changed to corner 4 and corner 4 changed to corner 1 is a symmetry of the tetrahedron.
This is not a symmetry, is it? Is everything correct so far? (Wondering)

Looks good.
Before we decide whether $\pi_5$ is a symmetry or not, perhaps we should list all symmetries of the tetrahedron? (Thinking)

Klaas van Aarsen said:
Looks good.
Before we decide whether $\pi_5$ is a symmetry or not, perhaps we should list all symmetries of the tetrahedron? (Thinking)

We have the identity.
We have rotations when 1 vertex is fixed.
We have rotations when no vertex is fixed.
We have reflections.

(Wondering)

mathmari said:
We have the identity.
We have rotations when 1 vertex is fixed.
We have rotations when no vertex is fixed.
We have reflections.

How many of each? (Wondering)

There are $4!=24$ permutations of 4 numbers.
The symmetries of the tetrahedron must be a subset with a number of elements that must divide 24. (Thinking)

Klaas van Aarsen said:
How many of each? (Wondering)

There are $4!=24$ permutations of 4 numbers.
The symmetries of the tetrahedron must be a subset with a number of elements that must divide 24. (Thinking)

We have 1 identity.

Rotations when one vertex is fixed: 8

So far we have 9.

To find the others, do we have to find all permitations of 4 numbers?

mathmari said:
We have 1 identity.

Rotations when one vertex is fixed: 8

So far we have 9.

To find the others, do we have to find all permitations of 4 numbers?

We can use the permutations to find more symmetries.
After all we already know that (1 2) corresponds to a reflection. Just like all other cycles of 2 numbers.
They are the reflections in the planes that contain an edge of the polyhedron.
How many would there be? (Wondering)

Klaas van Aarsen said:
We can use the permutations to find more symmetries.
After all we already know that (1 2) corresponds to a reflection. Just like all other cycles of 2 numbers.
They are the reflections in the planes that contain an edge of the polyhedron.
How many would there be? (Wondering)

We have (1 2), (1 3), (1 4), (2 3), (2 4), (3 4). So the number of such reflections is 6, right? (Wondering)

mathmari said:
We have (1 2), (1 3), (1 4), (2 3), (2 4), (3 4). So the number of such reflections is 6, right?

Yep. (Thinking)

Klaas van Aarsen said:
Yep. (Thinking)

So we have 6 more symmetries, so 9+6 = 15.

So are just the rotations missing? Or have I forgotten something?

mathmari said:
So we have 6 more symmetries, so 9+6 = 15.

So are just the rotations missing? Or have I forgotten something?

How about an axis of rotation through 2 opposing edges? (Wondering)

Which permutations of the 24 are left? (Wondering)

Klaas van Aarsen said:
How about an axis of rotation through 2 opposing edges? (Wondering)

Which permutations of the 24 are left? (Wondering)

We have the following:
• Identity. - 1
• Rotation so that one vertex is fixed. For each vertex there are two such rotations. Therefore is no total there 2* 4 = 8 such rotations. - 8
• Rotations where two pair of vertices are switched. This is a rotation of $180^{\circ}$ around the perpendicular bisector of the edges of the two pairs of vertices. These rotations are of the form: (1 2)(3 4), (1 3)(2 4), (1 4)(2 4). - 3
• Reflections about the plane that goes through one edge and the midpoint of the opposite edge. These reflections switch two edges. These reflections are of the form: (1 2), (1 3), (1 4), (2 3), (2 4), (3 4). - 6

What am I missing? (Wondering)

mathmari said:
We have the following:
• Identity. - 1
• Rotation so that one vertex is fixed. For each vertex there are two such rotations. Therefore is no total there 2* 4 = 8 such rotations. - 8
• Rotations where two pair of vertices are switched. This is a rotation of $180^{\circ}$ around the perpendicular bisector of the edges of the two pairs of vertices. These rotations are of the form: (1 2)(3 4), (1 3)(2 4), (1 4)(2 4). - 3
• Reflections about the plane that goes through one edge and the midpoint of the opposite edge. These reflections switch two edges. These reflections are of the form: (1 2), (1 3), (1 4), (2 3), (2 4), (3 4). - 6

What am I missing?

Which symmetries did we have for a square? (Wondering)

Klaas van Aarsen said:
Which symmetries did we have for a square? (Wondering)

At a square we have the rotations by 0, 90, 180, 270 degrees.
The reflections along the perpendicular bisectors.
The reflections along the diagonal and the antidiagonal.

Have I missed something?

mathmari said:
At a square we have the rotations by 0, 90, 180, 270 degrees.
The reflections along the perpendicular bisectors.
The reflections along the diagonal and the antidiagonal.

Have I missed something?

Ah, I actually wanted to point out that we can have compositions of a rotation and a reflection.
However, in the case of a square, these are also reflections. (Blush)

Either way, in 3 dimensions we have something called improper rotations, also called rotation-reflections.
For instance the permutation (1 2 3)(3 4) = (1 2 3 4) is a reflection followed by a rotation, which is also a symmetry in 3 dimensions. (Thinking)

Klaas van Aarsen said:
Either way, in 3 dimensions we have something called improper rotations, also called rotation-reflections.
For instance the permutation (1 2 3)(3 4) = (1 2 3 4) is a reflection followed by a rotation, which is also a symmetry in 3 dimensions. (Thinking)

Ahhh (Thinking)

At at the first question in each there is a symmetry of the tetrahedron that the permutation πi induces on the set of vertices {1,2,3,4}, right?

At the second question do we have to proof that the symmetry group of the tetraeder is the permutation group of the 4 numbers? Or which is the difference between the two questions? (Wondering)

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mathmari said:
Ahhh (Thinking)

At at the first question in each there is a symmetry of the tetrahedron that the permutation πi induces on the set of vertices {1,2,3,4}, right?

Yes. (Nod)
We found that $\pi_5 = (1\,2\,3\,4)$ is also a symmetry, and that it is geometrically a rotation-reflection.

mathmari said:
At the second question do we have to proof that the symmetry group of the tetraeder is the permutation group of the 4 numbers? Or which is the difference between the two questions? (Wondering)

In the first question we verified that 5 specific permutations of 4 numbers induced symmetries of the tetrahedron.
And we also specified which geometrical transformations they were.

In the second question we have to verify that all 24 permutations correspond to a geometrical symmetry of the tetrahedron.
You've listed 18 of them. We could already tell that it couldn't be right, since 18 does not divide 24.
Do we have all 24 now? (Wondering)

Klaas van Aarsen said:
In the first question we verified that 5 specific permutations of 4 numbers induced symmetries of the tetrahedron.
And we also specified which geometrical transformations they were.

In the second question we have to verify that all 24 permutations correspond to a geometrical symmetry of the tetrahedron.
You've listed 18 of them. We could already tell that it couldn't be right, since 18 does not divide 24.
Do we have all 24 now? (Wondering)

We have the following permuatations:

• Identity. - 1
• Rotations that one vertex is fixed. Such rotations are of the form: $(3\ \ 4 \ \ 2)$, $(4\ \ 2 \ \ 3)$, $(3\ \ 4 \ \ 1)$, $(4\ \ 1 \ \ 3)$, $(2\ \ 4 \ \ 1)$, $(4\ \ 1 \ \ 2)$, $(2\ \ 3 \ \ 1)$, $(3\ \ 1 \ \ 2)$. - 8
• Rotations that two pair of vertices are switched. Such rotations are of the form: $(1 2)(3 4)$, $(1 3)(2 4)$, $(1 4)(2 4)$. - 3
• Reflections that switch two vertices. Such reflections are of the form: $(1 2)$, $(1 3)$, $(1 4)$, $(2 3)$, $(2 4)$, $(3 4)$. - 6
• Rotations-reflections. For that do we take every product of a reflection with every rotation? Which order do we take? Or is it the same and we can consider both orders?
We would have:
$(3\ \ 4 \ \ 2)(1 2)$, $(4\ \ 2 \ \ 3)(1 2)$, $(3\ \ 4 \ \ 1)(1 2)$, $(4\ \ 1 \ \ 3)(1 2)$, $(2\ \ 4 \ \ 1)(1 2)$, $(4\ \ 1 \ \ 2)(1 2)$, $(2\ \ 3 \ \ 1)(1 2)$, $(3\ \ 1 \ \ 2)(1 2)$
$(3\ \ 4 \ \ 2)(1 3)$, $(4\ \ 2 \ \ 3)(1 3)$, $(3\ \ 4 \ \ 1)(1 3)$, $(4\ \ 1 \ \ 3)(1 3)$, $(2\ \ 4 \ \ 1)(1 3)$, $(4\ \ 1 \ \ 2)(1 3)$, $(2\ \ 3 \ \ 1)(1 3)$, $(3\ \ 1 \ \ 2)(1 3)$
$(3\ \ 4 \ \ 2)(1 4)$, $(4\ \ 2 \ \ 3)(1 4)$, $(3\ \ 4 \ \ 1)(1 4)$, $(4\ \ 1 \ \ 3)(1 4)$, $(2\ \ 4 \ \ 1)(1 4)$, $(4\ \ 1 \ \ 2)(1 4)$, $(2\ \ 3 \ \ 1)(1 4)$, $(3\ \ 1 \ \ 2)(1 4)$
$(3\ \ 4 \ \ 2)(2 3)$, $(4\ \ 2 \ \ 3)(2 3)$, $(3\ \ 4 \ \ 1)(2 3)$, $(4\ \ 1 \ \ 3)(2 3)$, $(2\ \ 4 \ \ 1)(2 3)$, $(4\ \ 1 \ \ 2)(2 3)$, $(2\ \ 3 \ \ 1)(2 3)$, $(3\ \ 1 \ \ 2)(2 3)$
$(3\ \ 4 \ \ 2)(2 4)$, $(4\ \ 2 \ \ 3)(2 4)$, $(3\ \ 4 \ \ 1)(2 4)$, $(4\ \ 1 \ \ 3)(2 4)$, $(2\ \ 4 \ \ 1)(2 4)$, $(4\ \ 1 \ \ 2)(2 4)$, $(2\ \ 3 \ \ 1)(2 4)$, $(3\ \ 1 \ \ 2)(2 4)$
$(3\ \ 4 \ \ 2)(3 4)$, $(4\ \ 2 \ \ 3)(3 4)$, $(3\ \ 4 \ \ 1)(3 4)$, $(4\ \ 1 \ \ 3)(3 4)$, $(2\ \ 4 \ \ 1)(3 4)$, $(4\ \ 1 \ \ 2)(3 4)$, $(2\ \ 3 \ \ 1)(3 4)$, $(3\ \ 1 \ \ 2)(3 4)$
If we calculate them we may see that we have some elements more than once? (Wondering)

Each of those must map to one of the possible 24 permutations of 4 numbers.
And not all of them will actually be rotation-reflections.
Which permutations are still missing such as (1234)? (Wondering)

Klaas van Aarsen said:
Which permutations are still missing such as (1234)? (Wondering)

The permutations that are missing are:
$(1 \ \ 4 \ \ 3 \ \ 2)$
$(1 \ \ 3 \ \ 4 \ \ 2)$
$(1 \ \ 2 \ \ 4 \ \ 3)$
$(1 \ \ 3 \ \ 2 \ \ 4)$
$(1 \ \ 4 \ \ 2 \ \ 3)$

Which am I still missing? (Wondering)

mathmari said:
The permutations that are missing are:
$(1 \ \ 4 \ \ 3 \ \ 2)$
$(1 \ \ 3 \ \ 4 \ \ 2)$
$(1 \ \ 2 \ \ 4 \ \ 3)$
$(1 \ \ 3 \ \ 2 \ \ 4)$
$(1 \ \ 4 \ \ 2 \ \ 3)$

Which am I still missing?

So there are 6 of them.
That makes 24 with the 18 you already have.
So we have all of them! (Happy)

Klaas van Aarsen said:
So there are 6 of them.
That makes 24 with the 18 you already have.
So we have all of them! (Happy)

Ahh yes!

So to show that all these permutations correspond to geometric symmetries we have to show that these 6 are rotation-reflections, since we have already seen the others, right?

So show this do we have to write them as product of smaller cycles? (Wondering)

mathmari said:
Ahh yes!

So to show that all these permutations correspond to geometric symmetries we have to show that these 6 are rotation-reflections, since we have already seen the others, right?

So show this do we have to write them as product of smaller cycles?

Yes. (Nod)

## 1. What is a permutation?

A permutation is a way of arranging or ordering a set of objects or elements in a specific order. It is essentially a rearrangement of the elements in the set.

## 2. How does a permutation induce on a set?

A permutation induces on a set by changing the order of the elements in the set according to the specific arrangement or ordering of the permutation. This means that the elements in the set will be rearranged in a different order than they were originally.

## 3. What is the difference between a permutation and a combination?

A permutation is an ordered arrangement of a set of elements, while a combination is an unordered selection of elements from a set. Permutations take into account the order of the elements, while combinations do not.

## 4. Can a permutation induce on a set with repeated elements?

Yes, a permutation can induce on a set with repeated elements. In this case, the permutation will rearrange the repeated elements in the same way, but the overall order of the set will still be different from the original order.

## 5. How is the number of possible permutations calculated?

The number of possible permutations for a set of n elements is calculated by using the factorial function, n!, which is equal to n x (n-1) x (n-2) x ... x 3 x 2 x 1. This means that for a set of 5 elements, there are 5! = 120 possible permutations.

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