# Sylow theorem and tetrahedron

According to Sylow's theorem, the group of isomorphism (rotations and mirror symmetries) of tetrahedron has a subgroup of order 8. How does one find it？ Moreover, is there a method to find a Sylow p-subgroup, or a subgroup of any order, of any regular polyhedron?

A way of seeing the Sylow $$2$$-subgroup of the symmetries of the tetrahedron comes from using the circumscribed cube such as in this animation. You get three order two orientation preserving symmetries by rotating the cube 180 degrees about any of the three coordinate axes. Together with the identity, these give a Klein four group.

Then we need a non-orientation preservin symmetry of order two to double that. You can see one also from that animation. Imagine a plane orthogonal to the top face of the cube that contains two opposite vertices of that top square. It is easy to convince yourself of the fact that this is also symmetry of the circumscribed tetrahedron.

With the coordinate axes parallel to the edges of that cube, and the vertices of the tetrahedron at the points $$(\pm1,\pm1,\pm1)$$ with any even number (zero or two) of minus signs), the copy of Klein four corresponds to diagonal metrices with $$\pm1$$ entries, again an even number of $$-1$$s. One of the reflections is then the linear transformation $$(x,y,z)\mapsto (y,x,z)$$.

• In my haste I forgot to describe the actual 2-subgroups. There are three of them. They all contain the described copy of Klein four. The choice of the remaining generator matters. Any 2-cycle of coordinates can be used. Nov 10, 2019 at 18:26

Consider the set $$E$$ of pairs of opposite edges. Its cardinal is three, and the group $$G$$ of automorphisms of the tetrahedron acts transitively on it. Consider the stabilizer $$H$$ of an element of $$E$$. It is of order $$24/3 = 8$$.

A geometric description could be given as follows: consider two opposite edges $$e_1$$ and $$e_2$$. There is a unique symmetry $$s_1$$ fixing $$e_2$$ and reversing $$e_1$$; there is a unique symmetry $$s_2$$ fixing $$e_1$$ and reversing $$e_2$$; there are symmetries exchanging $$e_1$$ and $$e_2$$. The group generated by these symmetries is the stabilizer of $$\{e_1,e_2\}$$. Continuing on this, one could show that it is dihedral.

Finally, the other $$2$$-Sylow can be found by considering the other two pairs of opposite edges.

EDIT: A general suggestion, for problems of this kind, would be the following. Assume $$G$$ is given as the group of automorphisms of some geometrical object $$O$$, and that you want to find a subgroup of $$G$$ of cardinal $$m$$. Then look for a set of $$m$$ things on $$O$$ that are preserved under the action of $$G$$ and such that the induced action is transitive; now just consider the stabilizer of one of these things.

• This is great. Better than what I discussed. Nov 10, 2019 at 18:26
• @Jyrki: Thanks :)
– Plop
Nov 11, 2019 at 16:10