generalized quaternion group
The groups given by the presentation
are the generalized quaternion groups. Generally one insists that as the properties of generalized quaternions become more uniform at this stage. However if then one observes so . Dihedral group properties are strongly related to generalized quaternion group properties because of their highly related presentations. We will see this in many of our results.
Given the relation (rather treating it as ) then as with dihedral groups we can shuffle words in to group all the at the beginning and the at the end. So every word takes the form . As and we have and . However we have an added relation that so we can write and also so we restrict to . This gives us elements of this form which makes the order of at most .
These relations are responsible for many useful results such as defining cross products for three-dimensional manipulations, and are also responsible for the most common example of a division ring. As a group, is a curious specimen of a -group in that it has only normal subgroups yet is non-abelian, it has a unique minimal subgroup and cannot be represented faithfully except by a regular representation – thus requiring degree 8. [To see this note that the unique minmal subgroup is necessarily normal, thus if a proper subgroup is the stabilizer of an action, then the minimal normal subgroup is in the kernel so the representation is not faithful.]
A common work around is to use matrices over but to treat these as matrices over .
A worthwhile additional example is . For this produces a group order 12 which is often overlooked.
2 Subgroup structure
As , then if is Hamiltonian then we require to be as well. However when we know has non-normal subgroups, for example . So we require . If then is cyclic and so trivially Hamiltonian. When we have the usual quaternion group of order 8 which is Hamiltonian by direct inspection: the conjugacy classes are , , , and , more commonly described by , , , and . In any case, all subgroups are normal. ∎
By way of converse it can be shown that the only finite Hamiltonian groups are where is abelian without an element of order 4. One sees already in that the subgroup is conjugate to the distinct subgroup and so such groups are not Hamiltonian.
for and for all .
Every subgroup of is either cyclic or a generalized quaternion.
The normal subgroups of are either subgroups of or and it is maximal subgroups (of index 2) of which there are 2 acyclic ones.
The order of elements of follows from standard cyclic group theory. Now for we simply compute: . So .
Now let be a subgroup of . If then is a subgroup of . We know the subgroups of are either cyclic or dihedral. If is cyclic then is cyclic (indeed it is a subgroup of or ). So assume that is dihedral. Then we have a dihedral presentation for . Now pullback this presentation to and we find is quaternion.
Finally, if does not contain then does not contain an element of the form , so and so it is cyclic.
For the normal subgroup structure, from the relation we see is normal. Thus all subgroups of are normal as is a normal cyclic subgroup. Next suppose is a normal subgroup not contained in . Then contains some , and so contains . Thus is a normal subgroup of . We know this forces to be contained in , a contradiction on our assumptions on , or and is a maximal subgroup (of index 2). ∎
has a unique minimal subgroup if and only if .
If and then has order and so the subgroup is of order , so it is minimal. As the center is also a minimal subgroup of order 2, then we do not have a unique minimal subgroup in these conditions. Thus .
Now suppose then is a -group so the minimal subgroups must all be of order 2. So we locate the elements of order 2. We have shown for any , and furthermore that . The only other minimal subgroups will be generated by for some , and as there is a unique minimal subgroup. ∎
|Title||generalized quaternion group|
|Date of creation||2013-03-22 16:27:41|
|Last modified on||2013-03-22 16:27:41|
|Last modified by||Algeboy (12884)|