Quaternion group

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Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). For example, the cycle in red reflects the fact that i 2 = −1, i 3 = −i  and i 4 = 1. The red cycle also reflects the fact that (−i )2 = −1, (−i )3 = i  and (−i )4 = 1.

In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication. It is often denoted by Q or Q8, and is given by the group presentation

where 1 is the identity element and −1 commutes with the other elements of the group.

Graphs

The Q8 group has the same order as the Dihedral group, D4, but a different structure, as shown by their Cayley graphs:

Cayley graph Cycle graphs
Cayley graph Q8.svg
Q8
The red arrows represent multiplication on the right by i, and the green arrows represent multiplication on the right by j.
Dih 4 Cayley Graph; generators a, b; prefix.svg
Dih4
Dihedral group
GroupDiagramMiniQ8.svg
Q8
Dih4 cycle graph.svg
Dih4

Cayley table

The Cayley table (multiplication table) for Q is given by:[1]

1 −1 i −i j −j k −k
1 1 −1 i −i j −j k −k
−1 −1 1 −i i −j j −k k
i i −i −1 1 k −k −j j
−i −i i 1 −1 −k k j −j
j j −j −k k −1 1 i −i
−j −j j k −k 1 −1 −i i
k k −k j −j −i i −1 1
−k −k k −j j i −i 1 −1

The multiplication of pairs of elements from the subset {±i, ±j, ±k} works like the cross product of unit vectors in three-dimensional Euclidean space.

Properties

The quaternion group has the unusual property of being Hamiltonian: every subgroup of Q is a normal subgroup, but the group is non-abelian.[2] Every Hamiltonian group contains a copy of Q.[3]

In abstract algebra, one can construct a real four-dimensional vector space with basis {1, i, j, k} and turn it into an associative algebra by using the above multiplication table and distributivity. The result is a skew field called the quaternions. Note that this is not quite the same as the group algebra on Q (which would be eight-dimensional). Conversely, one can start with the quaternions and define the quaternion group as the multiplicative subgroup consisting of the eight elements {1, −1, i, −i, j, −j, k, −k}. The complex four-dimensional vector space on the same basis is called the algebra of biquaternions.

Note that i, j, and k all have order four in Q and any two of them generate the entire group. Another presentation of Q[4] demonstrating this is:

One may take, for instance, i = x, j = y and k = x y.

The center and the commutator subgroup of Q is the subgroup {±1}. The factor group Q/{±1} is isomorphic to the Klein four-group V. The inner automorphism group of Q is isomorphic to Q modulo its center, and is therefore also isomorphic to the Klein four-group. The full automorphism group of Q is isomorphic to S4, the symmetric group on four letters. The outer automorphism group of Q is then S4/V which is isomorphic to S3.

Matrix representations

Q. g. as a subgroup of SL(2,C)

The quaternion group can be represented as a subgroup of the general linear group GL2(C). A representation

is given by

Since all of the above matrices have unit determinant, this is a representation of Q in the special linear group SL2(C). The standard identities for quaternion multiplication can be verified using the usual laws of matrix multiplication in GL2(C).[5]

Q. g. as a subgroup of SL(2,3)

There is also an important action of Q on the eight nonzero elements of the 2-dimensional vector space over the finite field F3. A representation

is given by

where {−1,0,1} are the three elements of F3. Since all of the above matrices have unit determinant over F3, this is a representation of Q in the special linear group SL(2, 3). Indeed, the group SL(2, 3) has order 24, and Q is a normal subgroup of SL(2, 3) of index 3.

Galois group

As Richard Dean showed in 1981, the quaternion group can be presented as the Galois group Gal(T/Q) where Q is the field of rational numbers and T is the splitting field, over Q, of the polynomial

.

The development uses the fundamental theorem of Galois theory in specifying four intermediate fields between Q and T and their Galois groups, as well as two theorems on cyclic extension of degree four over a field.[6]

Generalized quaternion group

A group is called a generalized quaternion group or dicyclic group if it has a presentation[4]

for some integer n ≥ 2. This group is denoted Q4n and has order 4n.[7] Coxeter labels these dicyclic groups <2,2,n>, being a special case of the binary polyhedral group <l,m,n> and related to the polyhedral groups (p,q,r), and dihedral group (2,2,n). The usual quaternion group corresponds to the case n = 2. The generalized quaternion group can be realized as the subgroup of GL2(C) generated by

where ωn = eiπ/n.[4] It can also be realized as the subgroup of unit quaternions generated by[8] x = eiπ/n and y = j.

The generalized quaternion groups have the property that every abelian subgroup is cyclic.[9] It can be shown that a finite p-group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above.[10] Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or generalized quaternion (of order a power of 2).[11] In particular, for a finite field F with odd characteristic, the 2-Sylow subgroup of SL2(F) is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group, Template:Harv. Letting pr be the size of F, where p is prime, the size of the 2-Sylow subgroup of SL2(F) is 2n, where n = ord2(p2 − 1) + ord2(r).

The Brauer–Suzuki theorem shows that groups whose Sylow 2-subgroups are generalized quaternion cannot be simple.

See also

Notes

  1. See also a table from Wolfram Alpha
  2. See Hall (1999), p. 190
  3. See Kurosh (1979), p. 67
  4. 4.0 4.1 4.2 Template:Harvnb
  5. Template:Harvnb
  6. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  7. Some authors (e.g., Template:Harvnb, pp. 87, 351) refer to this group as the dicyclic group, reserving the name generalized quaternion group to the case where n is a power of 2.
  8. Template:Harvnb, p. 98
  9. Template:Harvnb, p. 101, exercise 1
  10. Template:Harvnb, Theorem 11.6, p. 262
  11. Template:Harvnb, Theorem 4.3, p. 99

References

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  • Dean, Richard A. (1981) "A rational polynomial whose group is the quaternions", American Mathematical Monthly 88:42–5.
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External links