Quaternion group
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 Q_{8}, 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 Q_{8} group has the same order as the Dihedral group, D_{4}, but a different structure, as shown by their Cayley graphs:
Cayley graph | Cycle graphs | ||
---|---|---|---|
Q_{8} The red arrows represent multiplication on the right by i, and the green arrows represent multiplication on the right by j. |
Dih_{4} Dihedral group |
Q_{8} |
Dih_{4} |
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 S_{4}, the symmetric group on four letters. The outer automorphism group of Q is then S_{4}/V which is isomorphic to S_{3}.
Matrix representations
The quaternion group can be represented as a subgroup of the general linear group GL_{2}(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 SL_{2}(C). The standard identities for quaternion multiplication can be verified using the usual laws of matrix multiplication in GL_{2}(C).^{[5]}
There is also an important action of Q on the eight nonzero elements of the 2-dimensional vector space over the finite field F_{3}. A representation
is given by
where {−1,0,1} are the three elements of F_{3}. Since all of the above matrices have unit determinant over F_{3}, 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 Q_{4n} 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 GL_{2}(C) generated by
where ω_{n} = e^{iπ/n}.^{[4]} It can also be realized as the subgroup of unit quaternions generated by^{[8]} x = e^{iπ/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 SL_{2}(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 p^{r} be the size of F, where p is prime, the size of the 2-Sylow subgroup of SL_{2}(F) is 2^{n}, where n = ord_{2}(p^{2} − 1) + ord_{2}(r).
The Brauer–Suzuki theorem shows that groups whose Sylow 2-subgroups are generalized quaternion cannot be simple.
See also
- binary tetrahedral group
- Clifford algebra
- dicyclic group
- Hurwitz integral quaternion
- List of small groups
- 16-cell
Notes
- ↑ See also a table from Wolfram Alpha
- ↑ See Hall (1999), p. 190
- ↑ See Kurosh (1979), p. 67
- ↑ ^{4.0} ^{4.1} ^{4.2} Template:Harvnb
- ↑ Template:Harvnb
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ 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.
- ↑ Template:Harvnb, p. 98
- ↑ Template:Harvnb, p. 101, exercise 1
- ↑ Template:Harvnb, Theorem 11.6, p. 262
- ↑ 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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- P.R. Girard (1984) "The quaternion group and modern physics", European Journal of Physics 5:25–32.
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