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Definition by presentation
The quaternion group has the following presentation:
The identity is denoted , the common element is denoted , and the elements are denoted respectively.
The quaternion group is a group with eight elements, which can be described in any of the following ways:
- It is the holomorph of the ring .
- It is the holomorph of the cyclic group of order 4.
- It is the group comprising eight elements where 1 is the identity element, and all the other elements are squareroots of , such that and further, (the remaining relations can be deduced from these).
- It is the dicyclic group with parameter 2, viz .
- It is the Fibonacci group .
- The construction of the quaternion group can be mimicked for other primes giving, in general, a non-Abelian group of order . The general construction involves taking a semidirect product of the cyclic group of order with a subgroup of order in the automorphism group, say the subgroup generated by the automorphism taking an element to its .
- The quaternion group also generalizes to the family of dicyclic groups (also known as binary dihedral groups) and also to the family of generalized quaternion groups (which are the dicyclic groups whose order is a power of 2).
- The quaternion group is part of a larger family of -groups called extraspecial groups. An extraspecial group is a group of prime power order whose center, commutator subgroup and Frattini subgroup coincide, and are all cyclic of prime order.
Position in classifications
|Type of classification||Name in that classification|
|GAP ID||(8,4), i.e., the 4th among the groups of order 8|
|Hall-Senior number||5 among groups of order 8|
Further information: Element structure of quaternion group
Conjugacy class structure
|Conjugacy class||Size of conjugacy class||Order of elements in conjugacy class||Centralizer of first element of class|
|2||4||, same as|
|2||4||-- same as|
|2||4||-- same as|
Automorphism class structure
|Equivalence class (orbit) under action of automorphisms||Size of equivalence class (orbit)||Number of conjugacy classes in it||Size of each conjugacy class||Order of elements|
Basic arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 8#Arithmetic functions
Arithmetic functions of an element-counting nature
Further information: element structure of quaternion group
Arithmetic functions of a subgroup-counting nature
Further information: subgroup structure of quaternion group
|number of subgroups||6|
|number of conjugacy classes of subgroups||6|
|number of normal subgroups||6||groups with same order and number of normal subgroups | groups with same number of normal subgroups|
|number of automorphism classes of subgroups||4|
Lists of numerical invariants
|conjugacy class sizes||are each conjugacy classes of non-central elements.|
|degrees of irreducible representations||See linear representation theory of quaternion group|
|orders of subgroups||See subgroup structure of quaternion group|
Want to compare and contrast group properties with other groups of the same order? Check out groups of order 8#Group properties
Further information: Subgroup structure of quaternion group
|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||--||--|
Subgroup-defining functions and associated quotient-defining functions
Automorphisms and endomorphisms
Further information: endomorphism structure of quaternion group
|Construct||Value||Order||Second part of GAP ID (if group)|
|automorphism group||symmetric group:S4||24||12|
|inner automorphism group||Klein four-group||4||2|
|outer automorphism group||symmetric group:S3||6||1|
|group of class-preserving automorphisms||Klein four-group||4||2|
|group of IA-automorphisms||Klein four-group||4||2|
|quotient of class-preserving automorphism group by inner automorphism group||trivial group||1||1|
|quotient of IA-automorphism group by inner automorphism group||trivial group||1||1|
|group of center-fixing automorphisms||symmetric group:S4||24||12|
|extended automorphism group||direct product of S4 and Z2||48||48|
|inner holomorph||inner holomorph of D8 ( and the quaternion group have the same holomorph)||32||49|
Smallest of its kind
- This is a non-abelian nilpotent group of smallest possible order, along with dihedral group:D8.
- This is a non-abelian Dedekind group (or Hamiltonian group) of smallest possible order. Dedekind means that every subgroup is normal.
Different from others of the same order
- It is the only non-abelian Dedekind group of its order.
- It is the only non-abelian T-group of its order.
- It is the only group of its order for which the rank (in the sense of the maximum possible rank of an abelian subgroup) is strictly smaller than the minimum size of generating set: For this group, the former is 1 and the latter is 2.
This finite group has order 8 and has ID 4 among the groups of order 8 in GAP's SmallGroup library. For context, there are 5 groups of order 8. It can thus be defined using GAP's SmallGroup function as:
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(8,4);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [8,4]
or just do:
to have GAP output the group ID, that we can then compare to what we want.
|Description||Functions used||Mathematical comment|
|SylowSubgroup(SL(2,3),2)||SylowSubgroup and SL||The -Sylow subgroup of special linear group:SL(2,3)|
|ExtraspecialGroup(2^3,'-')||ExtraspecialGroup||The extraspecial group of order and '-' type|
|SylowSubgroup(SL(2,5),2)||SylowSubgroup and SL||The -Sylow subgroup of special linear group:SL(2,5)|
Adjoint group structures for quaternion group, Algebra group structures for quaternion group, Element structure of quaternion group, Endomorphism structure of quaternion group, Extensions for nontrivial outer action of Z2 on Q8, Extensions for nontrivial outer action of Z4 on Q8, Extensions for trivial outer action of V4 on Q8, Extensions for trivial outer action of Z2 on Q8, Fusion systems for quaternion group, Group cohomology of quaternion group, Linear representation theory of quaternion group, Projective representation theory of quaternion group, Second cohomology group for trivial group action of Q8 on V4, Second cohomology group for trivial group action of Q8 on Z2, Second cohomology group for trivial group action of Q8 on Z4, Subgroup structure of quaternion group, Supercharacter theories for quaternion group, Supergroups of quaternion group