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:
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
|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||trivial subgroup||1||8||1||1||1||quaternion group||0|
|center of quaternion group||cyclic group:Z2||2||4||1||1||1||Klein four-group||1|
|cyclic maximal subgroups of quaternion group||
||cyclic group:Z4||4||2||3||1||3||cyclic group:Z2||1|
|whole group||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|
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 is congruent to modulo . Here, , 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|
Subgroup-defining functions and associated quotient-defining functions
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 . 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) 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 -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 , while it is termed a p-core-automorphism-invariant subgroup if it is invariant under all automorphisms in the -core of the automorphism group. We have:
Characteristic -automorphism-invariant -core-automorphism-invariant normal
In the case of the quaternion group, we have the following:
- The characteristic subgroups are the same as the -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 -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 -groups containing the quaternion group in which these are not normal.
- The -core-automorphism-invariant subgroups are the same as the normal subgroups, which are the same as all subgroups.
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.
Abelian subgroups of maximum order
There are three abelian subgroups of maximum order: the three cyclic normal subgroups generated by 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 , 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.