generalized quaternion group

The groups given by the presentationMathworldPlanetmathPlanetmathPlanetmath


are the generalized quaternion groups. Generally one insists that n>1 as the properties of generalized quaternions become more uniform at this stage. However if n=1 then one observes a=b2 so Q4n4. 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.
  1. 1.


  2. 2.

    Q4n is abelianMathworldPlanetmathPlanetmath if and only if n=1.

  3. 3.

    Every element xQ4n can be written uniquely as x=aibj where 0i<2n and j=0,1.

  4. 4.


  5. 5.



Given the relationMathworldPlanetmath b-1ab=a-1 (rather treating it as ab=ba-1) then as with dihedral groupsMathworldPlanetmath we can shuffle words in {a,b} to group all the as at the beginning and the bs at the end. So every word takes the form aibj. As |a|=2n and |b|=4 we have 0i<2n and 0j<4. However we have an added relation that an=b2 so we can write aib2=ai+2 and also aib3=ai+2b so we restrict to j=0,1. This gives us 4n elements of this form which makes the order of Q4n at most 4n.

As an=b2 it follows [an,aibj]=[an,bj]=[b2,bj]=1. So an is central. If we quotient by an then we have the presentation


which we recognize as the presentation of the dihedral group. Thus Q4n/anD2n. This prove the order of Q4n is exactly 4n. Moreover, given aibjZ(Q4n) we have


So we have i=n. So anbj=bj+2. Then 1=[bj+2,a] forces j=0,2. This means Z(Q4n)=an=b2. ∎

1 Examples

As mentioned, if n=1 then Q44. If n=2 then we have the usual quaternion groupMathworldPlanetmathPlanetmath Q8. Because of the genesis of quaternionsMathworldPlanetmath, this group is often denoted with i,j,k 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, Q8 is a curious specimen of a p-group in that it has only normal subgroupsMathworldPlanetmath yet is non-abelianMathworldPlanetmathPlanetmath, it has a unique minimalPlanetmathPlanetmath subgroupMathworldPlanetmathPlanetmath and cannot be represented faithfully except by a regular representationPlanetmathPlanetmath – thus requiring degree 8. [To see this note that the unique minmal subgroup is necessarily normal, thus if a proper subgroupMathworldPlanetmath is the stabilizerMathworldPlanetmath of an action, then the minimal normal subgroup is in the kernel so the representationPlanetmathPlanetmath is not faithful.]

A common work around is to use 2×2 matrices over but to treat these as matrices over .


A worthwhile additional example is n=3. For this produces a group order 12 which is often overlooked.

2 Subgroup structure

Proposition 2.

Q4n is HamiltonianPlanetmathPlanetmath – meaning all a non-abelian groupMathworldPlanetmath whose subgroups are normal – if and only if n=2.


As Q4n/Z(Q4n)D2n, then if Q4n is Hamiltonian then we require D2n to be as well. However when n>2 we know D2n has non-normal subgroups, for example ab. So we require n2. If n=1 then Q4n is cyclic and so trivially Hamiltonian. When n=2 we have the usual quaternion group of order 8 which is Hamiltonian by direct inspection: the conjugacy classesMathworldPlanetmathPlanetmath are {1}, {a2}, {a,a3}, {b,a2b} and {ab,a3b}, more commonly described by {1}, {-1}, {i,-i}, {j,-j} and {k,-k}. In any case, all subgroups are normal. ∎

By way of converseMathworldPlanetmath it can be shown that the only finite Hamiltonian groups are AQ8 where A is abelian without an element of order 4. One sees already in 4Q8 that the subgroup (1,i) is conjugate to the distinct subgroup (1,-i) and so such groups are not Hamiltonian.

Proposition 3.
  1. 1.

    |ai|=2n/i for 1<i2n and |aib|=4 for all i.

  2. 2.

    Every subgroup of Q4n is either cyclic or a generalized quaternion.

  3. 3.

    The normal subgroups of Q4n are either subgroups of a or n=2i and it is maximal subgroups (of index 2) of which there are 2 acyclic ones.


The order of elements of a follows from standard cyclic groupMathworldPlanetmath theory. Now for aib we simply compute: (aib)2=aibaib=aia-ib2=b2. So |aib|=4.

Now let H be a subgroup of Q4n. If Z(Q4n)H then H/Z(Q4n) is a subgroup of D2n. We know the subgroups of D2n are either cyclic or dihedral. If H/Z(Q4n) is cyclic then H is cyclic (indeed it is a subgroup of a or H=aib). So assume that H/Z(Q4n) is dihedral. Then we have a dihedral presentation x,y:xm=1,y2=1,y-1xy=x-1 for H/Z(Q4n). Now pullback this presentation to H and we find H is quaternion.

Finally, if H does not contain Z(Q4n) then H does not contain an element of the form aib, so Ha and so it is cyclic.

For the normal subgroup structureMathworldPlanetmath, from the relation b-1ab=a-1 we see a is normal. Thus all subgroups of a are normal as a is a normal cyclic subgroup. Next suppose H is a normal subgroup not contained in a. Then H contains some aib, and so H contains Z(Q4n). Thus H/Z(Q4n) is a normal subgroup of D2n. We know this forces H/Z(Q4n) to be contained in a/Z(Q4n), a contradictionMathworldPlanetmathPlanetmath on our assumptionsPlanetmathPlanetmath on H, or n=2i and H/Z(Q4n) is a maximal subgroup (of index 2). ∎

Proposition 4.

Q4n has a unique minimal subgroup if and only if n=2i.


If p|n and p>2 then a2n/p has order p and so the subgroup a2n/p is of order p, 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 n=2i.

Now suppose n=2i then Q4n is a 2-group so the minimal subgroups must all be of order 2. So we locate the elements of order 2. We have shown |aib|=4 for any i, and furthermore that (aib)2=b2=an. The only other minimal subgroups will be generated by ai for some i, and as |a|=2i+1 there is a unique minimal subgroup. ∎

It can also be shown that any finite groupMathworldPlanetmath with a unique minimal subgroup is either cyclic of prime power order, or Q4n for some n=2i. We note that these groups have only regularPlanetmathPlanetmathPlanetmath faithful representationsMathworldPlanetmath.

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