Dihedralizing the Quaternions

Abstract

In this paper, we will take the classic dihedral and quaternion groups and explore questions like “what if we replace $i=e^{2\pi i/4}$ in ${\mathrm{Q}}_8$ with a larger root of unity?” and “what if we add a reflection to ${\mathrm{Q}}_8$?” The delightful answers reveal lesser-known families like the dicyclic, diquaternion, semidihedral, and semiabelian groups, which come to life with visuals such as Cayley graphs, cycle graphs, and subgroup lattices.

MSC:

• 20D99

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 In general, a subgroup lattice collapsed in this manner need not be an actual lattice.

2 By “the same,” we mean that the subgroup lattices, where each edge $H\le K$ is weighted by the index $[H:K]$, are identical. In particular, this means that ${\mathrm{C}}_2$ and ${\mathrm{C}}_3$ have different subgroup lattices.

3 The dihedral group ${\mathrm{D}}_6$ also decomposes as a semidirect product three ways, and as a direct product.

4 In a 2002 paper [1], this number was reported to be 49,487,365,422, but a correction appeared in 2022 [2].

5 John Conway said that the human race will never know the exact number of groups of order 2048, but in [4] he writes that it exceeds the $1,\mathrm{}774,\mathrm{}274,\mathrm{}116,\mathrm{}992,\mathrm{}170$ exponent-2 class 2 groups, and that the true number shares the first three digits.

Additional information

Notes on contributors

Matthew Macauley

MATTHEW MACAULEY is an Associate Professor at Clemson University. His research interests include algebraic biology and combinatorics. He teaches abstract algebra at the undergraduate and graduate levels using hundreds of colorful pictures, including ones that appear in this article (online version for color). He makes all of these freely available on his webpage, and encourages anyone interested to get in touch.