Dihedral Competitors - Quantum Calculus

Every abelian group A can be either ‘dihadrated’ or “dicycled”. The technical term is to form the generalized dihedral group Dih(A) or to form the generalized dicyclic group Dic(A,y) with an involution y in A. Let’s leave the superflous “generalized” away as it is clear what we mean in each case. In both cases we chose an additional “imaginary” element x. In the dihedral case $x^2=1$ and $x a x = a^{-1}$ for every $a \in A$ . In the dicyclic case $x^2=y$ and $x a x^{-1} = a^{-1}$ for all $a \in A$ . These constructions look very similar but their groups are drastically different. The dihedral groups are all natural, while the dicyclic groups are all not natural. The dihedral groups are all reflection groups (generated by involutions in the group), while in the dicyclic case, all involutions are in A. The reason is that any element off of A must have the form $xa$ for some $a \in A$ . But then $(xa)^2 = y \in A$ . But the assumption was that y is not the identity.

Here is a cute observation: if a group G of cardinality not larger than the continuum is not natural, then it has a dihedral competitor.

In the non-abelian case, we know by the structure theorem that all non-natural groups must be dicyclic $Dic(A,y)$ . But we know that $Dih(A)$ is then a competitor. In the Abelian case we also have a structure theorem and know that not natural groups are neither 2-torsion groups nor 2-divisible groups. Lets for simplicity look at finite Abelian groups only, where non-natural groups have even order $latx n=2m$ but are not 2-torsion groups $C_2^n$ . In each case we can form a dihedral competitor $Dih(A)$ , where $A$ is a subgroup of order m.

An other remark concerns the number $\phi(G)$ , which is the minimum over the number of groups compatible with a metric $(G,d)$ , where the minimum is taken over all metrics that are compatible with G. For example, $\phi(\mathbb{Z}) = 2$ as $\mathbb{Z}$ is not natural and admits the dihedral competitor $D_{\infty}$ . Ie call it the competitor function. An other nice example is $Q_8$ , the quaternion group. It is $Dic(A,y)$ for $A=C_4$ , the group of units in $\mathbb{C}$ . We have $\phi(Q_8) =4$ as the competitors are $Q_8, D_4,C_4 \times C_2, C_2^3$ . Lets just remark (and write down elsewhere)

The competitor function takes all positive values both on the class of abelian groups as well as on the class of non-abelian groups.

There is a nice coincidence here with naturallity. As Hurwitz has realized first, number theory in the division algebra of the quaternion is only interesting if one enhances the Lipschitz integers to Hurwitz integers. The units then are geometrically the 24 cell, the most exciting Platonic solids in all dimensions, as it is unique. 1-dimensional Platonic spheres are regular polygons, 2-dimensional Platonic spheres are the 5 platonic solids “Tetrahedron,Cube, Octahedron, Dodecahedron,Icosahedron” identified by Plato already, the 3-dimensional Platonic spheres contain the analog of these 5 as well as the 24 cell, which is self dual. In higher dimensions, we then only have the simplices, cross polytopes and hypercubes. The 24 cell is the group of units in the quaternion integers. It is 2T, the binary tetrahedral group. Its derived subgroup is $Q_8$ , its second derived group is $C_2$ so that as a non-metabelian group it is natural. It is satisfying to have the Hurwitz integer units natural and the Lipschitz integers non-natural.

Author: oliverknill