Full tetrahedral group is isomorphic to S4
This article gives a proof/explanation of the equivalence of multiple definitions for the term symmetric group:S4
View a complete list of pages giving proofs of equivalence of definitions
Statement
Consider the full tetrahedral group. This is the group of all self-isometries of that send a particular regular tetrahedron to itself. The claim is that this group is isomorphic to symmetric group:S4.
Proof
Construction of the homomorphism
The regular tetrahedron has four vertices. Number these vertices 1,2,3,4.
Every element of the full tetrahedral group permutes the vertices of the regular tetrahedron among themselves. Moreover, composing elements of the group composes the actions on the vertices. Therefore, the full tetrahedral group has a group action on the set . Therefore, there is a homomorphism of groups from the full tetrahedral group to symmetric group:S4 defined by this action.
Injectivity
The kernel of the homomorphism comprises those isometries of that fix all four vertices of the regular tetrahedron. We will show that this is trivial
Since the four vertices of a regular tetrahedron are affinely independent in the three-dimensional space , they affinely generate all of
, i.e., every point of
is expressible as an affine linear combination of these. Thus, any affine linear automorphism of
that fixes these four points must be the identity map. Any self-isometry of
is an affine linear automorphism, so any slef-isometry of
that fixes all points is the identity map.
Surjectivity
The essential argument is that the roles of the four points are completely interchangeable at the conceptual level, so every permutation of these must arise from a self-isometry of .
There are two more explicit styles of proof:
- For every permutation, describe how it arises from an isometry.
- Show that all elements of a particular generating set of symmetric group:S4 arise from isometries. For instance, we can use that transpositions generate the finitary symmetric group and then just show that all the transpositions arise from isometries.
We will follow the former approach below, because it encompasses the latter and also provides more information about the group structure. Note that the even permutations are realized via orientation-preserving isometries (which in this case includes only rotations, due to Euler's theorem) and the odd permutations involve reflections.
Partition | Partition in grouped form | Verbal description of cycle type | Elements with the cycle type | Size of conjugacy class | Even or odd? | How it is obtained from an isometry |
---|---|---|---|---|---|---|
1 + 1 + 1 + 1 | 1 (4 times) | four cycles of size one each, i.e., four fixed points | ![]() |
1 | even | The identity map on ![]() |
2 + 1 + 1 | 2 (1 time), 1 (2 times) | one transposition (cycle of size two), two fixed points | ![]() ![]() ![]() ![]() ![]() ![]() |
6 | odd | The reflection about the plane that is the perpendicular bisector of the line segment joining the two vertices that we wish to transpose. |
2 + 2 | 2 (2 times) | double transposition: two cycles of size two | ![]() ![]() ![]() |
3 | even | The half-turn (rotation by angle of ![]() ![]() |
3 + 1 | 3 (1 time), 1 (1 time) | one 3-cycle, one fixed point | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | even | Rotation by an angle ![]() ![]() ![]() |
4 | 4 (1 time) | one 4-cycle, no fixed points | ![]() ![]() ![]() ![]() ![]() ![]() |
6 | odd | (complicated, we need to realize this by composing the explicitly described isometries for the other permutations) |