Quaternion group

In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
formula_1 of the quaternions under multiplication. It is given by the group presentation
where "e" is the identity element and commutes with the other elements of the group.
Another presentation of Q8 is
Compared to dihedral group.
The quaternion group Q8 has the same order as the dihedral group , but a different structure, as shown by their Cayley and cycle graphs:
In the diagrams for D4, the group elements are marked with their action on a letter F in the defining representation R2. The same cannot be done for Q8, since it has no faithful representation in R2 or R3. D4 can be realized as a subset of the split-quaternions in the same way that Q8 can be viewed as a subset of the quaternions.
Cayley table.
The Cayley table (multiplication table) for Q8 is given by:
Properties.
Note that "i", "j", and "k" all have order four in Q8 and any two of them generate the entire group. Another presentation of Q8 based in only two elements to skip this redundancy is:
One may take, for instance, formula_5, and formula_6.
The quaternion group has the unusual property of being Hamiltonian: Q8 is non-abelian, but every subgroup is normal. Every Hamiltonian group contains a copy of Q8.
The quaternion group Q8 and the dihedral group D4 are the two smallest examples of a nilpotent non-abelian group.
The center and the commutator subgroup of Q8 is the subgroup formula_7. The inner automorphism group of Q8 is given by the group modulo its center, i.e. the factor group Q8/{e,}, which is isomorphic to the Klein four-group V. The full automorphism group of Q8 is isomorphic to S4, the symmetric group on four letters (see "Matrix representations" below), and the outer automorphism group of Q8 is thus S4/V, which is isomorphic to S3.
The quaternion group Q8 has five conjugacy classes, { e }, { }, { i, }, { j, }, { k, }, and so five irreducible representations over the complex numbers, with dimensions 1,1,1,1,2:
Trivial representation
Sign representations with i,j,k-kernel: Q8 has three maximal normal subgroups: the cyclic subgroups generated by i, j, and k respectively. For each maximal normal subgroup "N", we obtain a one-dimensional representation factoring through the 2-element quotient group "G"/"N". The representation sends elements of "N" to 1, and elements outside "N" to -1.
2-dimensional representation: Described below in "Matrix representations".
The character table of Q8 turns out to be the same as that of D4: 
Since the irreducible characters formula_8 in the rows above have real values, this gives the decomposition of the real group algebra of formula_9 into minimal two-sided ideals: formula_10, where the idempotents formula_11 correspond to the irreducibles: formula_12, so thatformula_13
formula_14
formula_15
formula_16
formula_17.Each of these irreducible ideals is isomorphic to a real central simple algebra, the first four to the real field formula_18. The last ideal formula_19 is isomorphic to the skew field of quaternions formula_20 by the correspondence:formula_21Furthermore, the projection homomorphism formula_22 given by formula_23 has kernel ideal generated by the idempotent:formula_24 so the quaternions can also be obtained as the quotient ring formula_25.
The complex group algebra is thus formula_26, where formula_27 is the algebra of biquaternions.
Matrix representations.
The two-dimensional irreducible complex representation described above gives the quaternion group Q8 as a subgroup of the general linear group formula_28. The quaternion group is a multiplicative subgroup of the quaternion algebra formula_29, which has a regular representation formula_30 by left multiplication on itself considered as a complex vector space with basis formula_31, so that formula_32 corresponds to the C-linear mapping formula_33. The resulting representation formula_34 is given by:
Since all of the above matrices have unit determinant, this is a representation of Q8 in the special linear group SL2(C).
A variant gives a representation by unitary matrices (table at right). Let formula_36 correspond to the linear mapping formula_37, so that formula_38 is given by:
formula_39
There is also an important action of Q8 on the 2-dimensional vector space over the finite field F3 = {0,1,−1} (table at right). A modular representation formula_40 is given by
This representation can be obtained from the extension field F9 = F3["k"] = F31 + F3"k", where "k"2 = −1 and the multiplicative group (F9)× has generators ±("k"+1), ±("k"-1) of order 8. The two-dimensional F3-vector space F9 admits the linear mappings formula_42 for "z" in F9, as well as the Frobenius automorphism formula_43 satisfying formula_44 and formula_45. Then the above representation matrices are formula_46, formula_47, formula_48, and formula_49.
The above representation realizes Q8 as a normal subgroup of . Thus, for each matrix formula_50, we have a group automorphism formula_51 defined by formula_52, with formula_53. In fact, these give the full automorphism group as: formula_54, This is isomorphic to the symmetric group S4 since the linear mappings formula_55 permute the four one-dimensional subspaces of formula_56, i.e. the four points of the projective space formula_57.
Also, this representation permutes the eight non-zero vectors of (F3)2, giving an embedding of Q8 in the symmetric group S8, in addition to the embeddings given by the regular representations.
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.
Generalized quaternion group.
A generalized quaternion group Q4"n" of order 4"n" is defined by the presentation
for an integer , with the usual quaternion group given by "n" = 2. Coxeter calls Q4"n" the dicyclic group formula_60, a special case of the binary polyhedral group formula_61 and related to the polyhedral group formula_62 and the dihedral group formula_63. The generalized quaternion group can be realized as the subgroup of formula_28 generated by
where formula_66. It can also be realized as the subgroup of unit quaternions generated by formula_67 and formula_68.
The generalized quaternion groups have the property that every abelian subgroup is cyclic. 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. Another characterization is that a finite "p"-group in which there is a unique subgroup of order "p" is either cyclic or a 2-group isomorphic to generalized quaternion group. 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, . Letting "pr" be the size of "F", where "p" is prime, the size of the 2-Sylow subgroup of SL2("F") is 2"n", where .
The Brauer–Suzuki theorem shows that the groups whose Sylow 2-subgroups are generalized quaternion cannot be simple.
Another terminology reserves the name "generalized quaternion group" for a dicyclic group of order a power of 2, which admits the presentation