Subgroup structure of quaternion group

This article gives specific information, namely, subgroup structure, about a particular group, namely: quaternion group.
View subgroup structure of particular groups | View other specific information about quaternion group

The quaternion group is an 8-element group on the set:

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

See the multiplication table of the group at quaternion group and more on the element structure at element structure of quaternion group.

Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (group of prime power order)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so all subgroups have prime power orders)|order of quotient group divides order of group (and equals index of corresponding normal subgroup, so all quotients have prime power orders)
prime power order implies not centerless | prime power order implies nilpotent | prime power order implies center is normality-large
size of conjugacy class of subgroups divides index of center
congruence condition on number of subgroups of given prime power order: The total number of subgroups of any fixed prime power order is congruent to 1 mod the prime.

Tables classifying subgroups up to automorphisms

Note that all subgroups are normal subgroups, so the quotient group exists in all cases.

Automorphism class of subgroups	List of subgroups	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (=1 iff automorph-conjugate subgroup)	Size of each conjugacy class (=1 iff normal subgroup)	Total number of subgroups (=1 iff characteristic subgroup)	Isomorphism class of quotient (if exists)	Nilpotency class
trivial subgroup	$\{1\}$	trivial subgroup	1	8	1	1	1	quaternion group	0
center of quaternion group	$\{1,-1\}$	cyclic group:Z2	2	4	1	1	1	Klein four-group	1
cyclic maximal subgroups of quaternion group	$\{1,-1,i,-i\}$ $\{1,-1,j,-j\}$ $\{1,-1,k,-k\}$	cyclic group:Z4	4	2	3	1	3	cyclic group:Z2	1
whole group	$\{1,-1,i,-i,j,-j,k,-k\}$	quaternion group	8	1	1	1	1	trivial group	2
Total (4 rows)	--	--	--	--	6	--	6	--	--

Table classifying isomorphism types of subgroups

The first part of the GAP ID is the order of the subgroup.

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

Table classifying number of subgroups by order

Note that these orders satisfy the congruence condition on number of subgroups of given prime power order: the number of subgroups of order $p^r$ is congruent to $1$ modulo $p$. Here, $p=2$, so this means that the number of subgroups of any given order is odd.

Group order	Index	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Automorphism classes of occurrences as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
1	8	1	1	1	1	1
2	4	1	1	1	1	1
4	2	3	3	1	3	1
8	1	1	1	1	1	1
Total	--	6	6	4	6	3

Defining functions

Subgroup-defining functions and associated quotient-defining functions

Subgroup-defining function	What it means	Value as subgroup	Value as group	Order	Associated quotient-defining function	Value as group	Order (= index of subgroup)
center	elements that commute with every group element	center of quaternion group: $\{1,-1\}$	cyclic group:Z2	2	inner automorphism group	Klein four-group	4
derived subgroup	subgroup generated by all commutators	center of quaternion group: $\{1,-1\}$	cyclic group:Z2	2	abelianization	Klein four-group	4
Frattini subgroup	intersection of all maximal subgroups	center of quaternion group: $\{1,-1\}$	cyclic group:Z2	2	Frattini quotient	Klein four-group	4
Jacobson radical	intersection of all maximal normal subgroups	center of quaternion group: $\{1,-1\}$	cyclic group:Z2	2	?	Klein four-group	4
socle	join of all minimal normal subgroups	center of quaternion group: $\{1,-1\}$	cyclic group:Z2	2	?	Klein four-group	4
Fitting subgroup	join of all nilpotent normal subgroups	whole group	quaternion group	8	Fitting quotient	trivial group	1
join of abelian subgroups of maximum order	join of all abelian subgroups of maximum order among abelian subgroups	whole group	quaternion group	8	?	trivial group	1
join of abelian subgroups of maximum rank	join of all abelian subgroups of maximum rank among abelian subgroups	whole group	quaternion group	8	?	trivial group	1
join of elementary abelian subgroups of maximum order	join of all elementary abelian subgroups of maximum order among elementary abelian subgroups	center of quaternion group: $\{1,-1\}$	cyclic group:Z2	2	?	Klein four-group	4
ZJ-subgroup	center of the join of abelian subgroups of maximum order	center of quaternion group: $\{1,-1\}$	cyclic group:Z2	2	?	Klein four-group	4

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

Note that since all subgroups are normal, the lattice of subgroups coincides with the lattice of normal subgroups. 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.

This is in the Hall-Senior family (up to isocliny) ${\mathrm{\Gamma }}_2$ and has the Hall-Senior genus as the dihedral group:D8. The general picture of the lattice of normal subgroups of that Hall-Senior genus is given below:

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.

Maximality notions related to abelianness

Abelian subgroups of maximum order

There are three abelian subgroups of maximum order: the three cyclic normal subgroups generated by $i,j,k$ respectively. These are all automorphic subgroups. Together, they generate the whole group. These are also the only subgroups maximal among abelian subgroups.

In particular, the join of abelian subgroups of maximum order, sometimes called the Thompson subgroup and denoted by $J$, is the whole group. Thus, the ZJ-subgroup, which is defined as the center of this Thompson subgroup, equals the center of the whole group.

Abelian subgroups of maximum rank

The quaternion group has rank one: every abelian subgroup is cyclic. Thus, the abelian subgroups of maximum rank are the center and the three subgroups of order four. The join of these, i.e., the join of abelian subgroups of maximum rank, is thus the whole group.

Elementary abelian subgroups of maximum order

There is exactly one elementary abelian subgroup of maximum order: the center.