Generalized quaternion group: Difference between revisions

Latest revision as of 15:46, 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.

The generalized quaternion group is generally only ever defined for $k \geq 2$ . However, if we put $k = 1$ we retrieve the Klein four-group.

Property	Satisfied	Explanation
Abelian group	No
Nilpotent group	Yes. Nilpotency class $k$
Solvable group	Yes
Supersolvable group	Yes
Metacyclic group	Yes
Ambivalent group	Yes
Rational group	Yes only for $k = 2$ , i.e., 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]].
+The generalized quaternion group is generally only ever defined for <math>k \geq 2</math>. However, if we put <math>k=1</math> we retrieve the [[Klein four-group]].
+==Group properties==
+{| class="wikitable" border="1"
+!Property !! Satisfied !! Explanation
+|-
+|[[Abelian group]] || No ||
+|-
+|[[Nilpotent group]] || Yes. Nilpotency class <math>k</math> ||
+|-
+|[[Solvable group]] || Yes ||
+|-
+|[[Supersolvable group]] || Yes ||
+|-
+|[[Metacyclic group]] || Yes ||
+|-
+|[[Ambivalent group]] || Yes ||
+|-
+|[[Rational group]] || Yes only for <math>k =2</math>, i.e., the [[quaternion group]] ||
+|}
 ==Examples==