The permutation induces on the set

[mathmari]

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)

 

[I like Serena]

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)

 

[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)

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)

 

[I like Serena]

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

 

[mathmari]

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)

 

[I like Serena]

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)

 

[mathmari]

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?

 

[I like Serena]

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)

 

[mathmari]

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)

 

[I like Serena]

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)

 

[mathmari]

Yep. (Thinking)

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

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

 

[I like Serena]

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)

 

[mathmari]

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)

 

[I like Serena]

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)

 

[mathmari]

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?

 

[I like Serena]

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)

 

[mathmari]

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)

 

[I like Serena]

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.

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)

 

[mathmari]

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)

 

[I like Serena]

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.
For instance (123)(12)=(13), which is a reflection we already had.
Which permutations are still missing such as (1234)? (Wondering)

 

[mathmari]

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)

 

[I like Serena]

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)

 

[mathmari]

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)

 

[I like Serena]

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)