Subgroup structure of quaternion group

The quaternion group has four types of subgroups, when classified upto automorphism:

1. The trivial subgroup. (1)
2. The center, given by $\pm 1$. This is isomorphic to the cyclic group of order two. (1)
3. The three four-element cyclic subgroups, generated by the elements $i,j,k$ respectively. They are all related by outer automorphisms, though no two of them are conjugate. (In fact, they're all normal). Each is isomorphic to the cyclic group of order four. (3)
4. The whole group. (1)

Tables for quick information

Group name	GAP ID	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
Trivial group	$(1,1)$	1	1	1	1
Cyclic group:Z2	$(2,1)$	1	1	1	1
Cyclic group:Z4	$(4,1)$	3	3	3	0
Quaternion group	$(8,4)$	1	1	1	1
Total	--	6	6	6	3

The center (type (2))

This is a two-element subset given by $\pm 1$.

Subgroup-defining functions yielding this subgroup

• Omega-1: The ${\mathrm{\Omega }}_1$ subgroup is the center: it is the subgroup generated by all the elements of order two.
• Agemo-1: The ${\mho }^1$ subgroup is the center: it is the subgroup generated by all the squares.
• Center: This subset is the center.
• Commutator subgroup: This subset is the commutator subgroup.
• Frattini subgroup: This subset is the intersection of the three maximal subgroups, which are the subgroups generated by $i,j,k$ respectively.
• Socle: In fact, the center is the unique minimal normal subgroup.
• Jacobson radical: The center is the intersection of all maximal normal subgroups (for a group of prime power order, maximal subgroups are the same thing as maximal normal subgroups, so this is anyway the same as the Frattini subgroup).

Subgroup properties satisfied

On account of being an agemo subgroup, as well as on account of being the commutator subgroup, the center is a verbal subgroup. It thus satisfies the following properties:

• Fully characteristic subgroup
• Image-closed fully characteristic subgroup
• Characteristic subgroup
• Image-closed characteristic subgroup

On account of being an omega subgroup, the center is a variety-containing subgroup. In particular, it satisfies the following properties:

• Homomorph-containing subgroup
• Isomorph-free subgroup
• Intermediately fully characteristic subgroup
• Intermediately characteristic subgroup

On account of being the center, it is also a:

• Normal subgroup
• Central subgroup
• Normality-large subgroup: For full proof, refer: Nilpotent implies center is normality-large
• Central factor
• Hereditarily normal subgroup

The four-element subgroups (type (3))

All these subgroups are normal of order four, and they are all automorphic subgroups. The subgroups are explicitly:

$\{1,i,-1,-i\},\{1,j,-1,-j\},\{1,k,-1,-k\}$.

Subgroup properties satisfied by these subgroups

• Normal subgroup
• Maximal normal subgroup
• Contracharacteristic subgroup
• Isomorph-automorphic subgroup
• Order-automorphic subgroup
• Self-centralizing subgroup

Subgroup properties not satisfied by these subgroups

• Characteristic subgroup

Lattice of subgroups

The entire lattice

The lattice of subgroups of the quaternion group has the following interesting features:

• Since all subgroups are normal, but the group is not abelian, the inner automorphism group is a nontrivial group of automorphisms that fixes all elements of the lattice.
• The outer automorphism group is isomorphic to the symmetric group of degree three. This group fixes each of the three characteristic subgroups: the trivial subgroup, the whole group, and the two-element center $\{-1,1\}$. The three normal subgroups of order four are not characteristic and the elements of the outer automorphism group give rise to permutations on this set of subgroups.
• The lattice does not enjoy reverse symmetry, in the sense that it is not isomorphic to its reverse lattice. This is because there are three maximal subgroups while there is only one minimal subgroup.

The sublattice of normal subgroups

The lattice of normal subgroups is the same as the lattice of subgroups, because every subgroup is normal.

The lattice of normal subgroups of the quaternion group is isomorphic as a lattice to the lattice of normal subgroups of the dihedral group:D8. However, the lattice of all subgroups of the dihedral group is substantially bigger.

The sublattice of characteristic subgroups

The lattice of characteristic subgroups of the quaternion group is a totally ordered lattice with three elements: the trivial subgroup, the unique subgroup of order two, and the whole group. These subgroups are also fully characteristic, in fact verbal.

Aspects of subgroup structure relevant for embeddings in bigger groups

2-automorphism-invariance and 2-core-automorphism-invariance

A subgroup of a $p$-group is termed a p-automorphism-invariant subgroup if it is invariant under all automorphisms of the whole group whose order is a power of $p$, while it is termed a p-core-automorphism-invariant subgroup if it is invariant under all automorphisms in the $p$-core of the automorphism group. We have:

Characteristic $⟹$ $p$-automorphism-invariant $⟹$ $p$-core-automorphism-invariant $⟹$ normal

In the case of the quaternion group, we have the following:

• The characteristic subgroups are the same as the $2$-automorphism-invariant subgroups, namely: the whole group, the trivial subgroup, and the center. Thus, the only subgroups of the quaternion group that are normal in every $2$-group containing it are the whole group, the trivial subgroup, and the center. In other words, for each of the subgroups of order four, we can find bigger $2$-groups containing the quaternion group in which these are not normal.
• The $2$-core-automorphism-invariant subgroups are the same as the normal subgroups, which are the same as all subgroups.

Coprime automorphism-invariance

Further information: Coprime automorphism-invariant normal subgroup of Hall subgroup is normalizer-relatively normal, isomorph-normal coprime automorphism-invariant of Sylow implies weakly closed

The coprime automorphism-invariant subgroups are the same as the coprime automorphism-invariant normal subgroups, which are the same as the characteristic subgroups. In other words, these are only the trivial subgroup, the whole group, and the center. In particular, this means that for any of the subgroups of order four, we can find a bigger group in which the quaternion group is Sylow, but that particular subgroup is not a normalizer-relatively normal subgroup.