Generalized quaternion group: Difference between revisions

Revision as of 15:39, 15 December 2023

A generalized quaternion group is a group of order $2^{k + 1}$ with generators $x$ and $a$ such that the group has the presentation:

$< a, x | x^{2} = a^{2^{k - 1}}, a^{2^{k}} = 1, x a x^{- 1} = a^{- 1} >$

Equivalently, it is the dicyclic group with parameter $2^{k - 1}$ .

For the particular case $k = 2$ , we recover the quaternion group.

$k$	Group	Order, $2^{k + 1}$
2	quaternion group	8
3	generalized quaternion group:Q16	16
4	generalized quaternion group:Q32	32
5	generalized quaternion group:Q64	64
6	generalized quaternion group:Q128	128
7	generalized quaternion group:Q256	256
8	generalized quaternion group:Q512	512
9	generalized quaternion group:Q1024	1024
10	generalized quaternion group:Q2048	2048

@@ Line 8: / Line 8: @@
 For the particular case <math>k=2</math>, we recover the [[quaternion group]].
+===Small values===
+{| class="sortable" border="1"
+!<math>k</math> !! Group !! Order, <math>2^{k+1}</math>
+|-
+| 2 || [[quaternion group]] || 8
+|-
+| 3 || [[generalized quaternion group:Q16]] || 16
+|-
+| 4 || [[generalized quaternion group:Q32]] || 32
+|-
+| 5 || [[generalized quaternion group:Q64]] || 64
+|-
+| 6 || [[generalized quaternion group:Q128]] || 128
+|-
+| 7 || [[generalized quaternion group:Q256]] || 256
+|-
+| 8 || [[generalized quaternion group:Q512]] || 512
+|-
+| 9 || [[generalized quaternion group:Q1024]] || 1024
+|-
+| 10 || [[generalized quaternion group:Q2048]] || 2048
+|}