Generalized quaternion group: Difference between revisions

Revision as of 23:21, 22 September 2007

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

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

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

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

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 A '''generalized quaternion group''' is a group of order <math>2^{k+1}</math> with generators <math>x</math> and <math>a</math> such that the group has the presentation:
-<math><a,x| x^2 = a^{2^k} = 1, xax^{-1} = a^{-1}></math>
+<math><a,x| x^2 = a^{2^{k-1}}, a^{2^k} = 1, xax^{-1} = a^{-1}></math>
 Equivalently, it is the [[dicyclic group]] with parameter <math>2^{k-1}</math>.
 For the particular case <math>k=2</math>, we recover the [[quaternion group]].