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.
Proposition 1.
Proof.
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 .
1 Examples
As mentioned, if then . If then we have the
usual quaternion group . Because of the genesis of quaternions
, this group is often denoted with relations as follows:
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
Proposition 2.
is Hamiltonian – meaning all a non-abelian group
whose subgroups are normal – if and only
if .
Proof.
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.
Proposition 3.
-
1.
for and for all .
-
2.
Every subgroup of is either cyclic or a generalized quaternion.
-
3.
The normal subgroups of are either subgroups of or and it is maximal subgroups (of index 2) of which there are 2 acyclic ones.
Proof.
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).
∎
Proposition 4.
has a unique minimal subgroup if and only if .
Proof.
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. ∎
It can also be shown that any finite group with a unique minimal subgroup is either
cyclic of prime power order, or for some .
We note that these groups have only regular
faithful representations
.
Title | generalized quaternion group |
---|---|
Canonical name | GeneralizedQuaternionGroup |
Date of creation | 2013-03-22 16:27:41 |
Last modified on | 2013-03-22 16:27:41 |
Owner | Algeboy (12884) |
Last modified by | Algeboy (12884) |
Numerical id | 7 |
Author | Algeboy (12884) |
Entry type | Derivation |
Classification | msc 20A99 |
Synonym | quaternion groups |
Related topic | DihedralGroupProperties |
Defines | generalized quaternion |