What are all the elements of the group of symmetries of a regular tetrahedron?

One of them

enter image description here

Use different axes (connecting the midpoints of two opposite edges of the tetrahedron) to get the other products of two disjoint 2-cycles as 180 degree rotations.

Here is the same animation with the cube surrounding the tetrahedron shown as a wireframe. The axis of rotation joins the centers of two opposite faces of the cube

enter image description here


Consider the set of half edges of the tetrahedron. It is easy to convince yourself that the symmetry group of the solid allows you to move from one such half edge to any other in exactly one way. Since there are six edges, there are twelve half edges, and this implies that the group of symmetries has exactly 12 elements.

Now the group obviously permutes the four vertices. Since there is exactly one subgroup of order 12 in $S_4$, namely the alternating group, it must be our group.

Tags:

Permutations

Group Theory

Related

Is $1000000000000066600000000000001$ (Belphegor's prime) actually a prime? Prove that there are infinitely many integers n such that $4n^2+1$ is divisible by both $13$ and $5$ Show $x-1$ is irreducible in $\mathbb{Z}_8[x]$ Infinite groups whose non-trivial subgroups are of finite index Finding the integral $I=\int_0^1{x^{-2/3}(1-x)^{-1/3}}dx$ Is there a function for every possible line? Prove that $\vert E \vert=0$ if $\frac{x+y}{2}\notin E$ for $x,y \in E$ Joint PDF of two random variables in a triangle Determine the third point in right triangle only knowing the coordinates of the other two points proof - Show that $1! +2! +3!+\cdots+n!$ is a perfect power if and only if $n=3$ Can a complex number be prime? If a, b, c are three natural numbers with $\gcd(a,b,c) = 1$ such that $\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$ then show that $a+b$ is a square.

Recent Posts