Quaternion group
In group theory, the quaternion group Q_{8} (sometimes just denoted by Q) is a nonabelian group of order eight, isomorphic to a certain eightelement subset of the quaternions under multiplication. It is given by the group presentation
Algebraic structure → Group theory Group theory 



Infinite dimensional Lie group

where e is the identity element and e commutes with the other elements of the group.
Compared to dihedral group
The quaternion group Q_{8} has the same order as the dihedral group D_{4}, but a different structure, as shown by their Cayley and cycle graphs:
Q_{8}  D_{4}  

Cayley graph  Red arrows represent multiplication by i, green arrows by j. 

Cycle graph 
The dihedral group D_{4} can be realized as a subset of the splitquaternions in the same way that Q_{8} can be viewed as a subset of the quaternions.
Cayley table
The Cayley table (multiplication table) for Q_{8} is given by:[1]
×  e  e  i  i  j  j  k  k 

e  e  e  i  i  j  j  k  k 
e  e  e  i  i  j  j  k  k 
i  i  i  e  e  k  k  j  j 
i  i  i  e  e  k  k  j  j 
j  j  j  k  k  e  e  i  i 
j  j  j  k  k  e  e  i  i 
k  k  k  j  j  i  i  e  e 
k  k  k  j  j  i  i  e  e 
Properties
The quaternion group has the unusual property of being Hamiltonian: every subgroup of Q_{8} is a normal subgroup, but the group is nonabelian.[2] Every Hamiltonian group contains a copy of Q_{8}.[3]
The quaternion group Q_{8} is one of the two smallest examples of a nilpotent nonabelian group, the other being the dihedral group D_{4} of order 8.
The quaternion group Q_{8} has five irreducible representations, and their dimensions are 1,1,1,1,2. The proof for this property is not difficult, since the number of irreducible characters of Q_{8} is equal to the number of its conjugacy classes, which is five ( { e }, { e }, { i, i }, { j, j }, { k, k } ).
These five representations are as follows:
Trivial representation
Sign representations with i,j,kkernel: Q_{8} has three maximal normal subgroups: the cyclic subgroups generated by i, j, and k respectively. For each maximal normal subgroup, we obtain a onedimensional representation with that subgroup as kernel. The representation sends elements inside the subgroup to 1, and elements outside the subgroup to 1.
2dimensional representation: A representation : is given below in the Matrix representations section.
So the character table of the quaternion group Q_{8}, which turns out to be the same as the character table of the dihedral group D_{4}, is:
Representation/Conjugacy class  { e }  { e }  { i, i }  { j, j }  { k, k } 

Trivial representation  1  1  1  1  1 
Sign representations with ikernel  1  1  1  1  1 
Sign representations with jkernel  1  1  1  1  1 
Sign representations with kkernel  1  1  1  1  1 
2dimensional representation  2  2  0  0  0 
In abstract algebra, one can construct a real fourdimensional vector space as the quotient of the group ring R[Q] by the ideal defined by span_{R}({e+e, i+i, j+j, k+k}). The result is a skew field called the quaternions. Note that this is not quite the same as the group algebra on Q_{8} (which would be eightdimensional). 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 fourdimensional vector space on the same basis is called the algebra of biquaternions.
Note that i, j, and k all have order four in Q_{8} and any two of them generate the entire group. Another presentation of Q_{8}[4] demonstrating this is:
One may take, for instance, i = x, j = y and k = xy.
The center and the commutator subgroup of Q_{8} is the subgroup {e,e}. The factor group Q_{8}/{e,e} is isomorphic to the Klein fourgroup V. The inner automorphism group of Q_{8} is isomorphic to Q_{8} modulo its center, and is therefore also isomorphic to the Klein fourgroup. The full automorphism group of Q_{8} is isomorphic to the symmetric group of degree 4, S_{4}, the symmetric group on four letters. The outer automorphism group of Q_{8} 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_{8} 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_{8} on the eight nonzero elements of the 2dimensional 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_{8} in the special linear group SL(2, 3). Indeed, the group SL(2, 3) has order 24, and Q_{8} 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 generalized quaternion group[7] is a dicyclic group of order a power of 2.
It is a part of more general class of dicyclic groups.
Some authors define [4] generalized quaternion group to be the same as dicyclic group.
for some integer n ≥ 2. This group is denoted Q_{4n} and has order 4n.[8] 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[9] x = e^{iπ/n} and y = j.
The generalized quaternion groups have the property that every abelian subgroup is cyclic.[10] It can be shown that a finite pgroup with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above.[11] Another characterization is that a finite pgroup in which there is a unique subgroup of order p is either cyclic or a 2group isomorphic to generalized quaternion group.[12] In particular, for a finite field F with odd characteristic, the 2Sylow subgroup of SL_{2}(F) is nonabelian and has only one subgroup of order 2, so this 2Sylow subgroup must be a generalized quaternion group, (Gorenstein 1980, p. 42). Letting p^{r} be the size of F, where p is prime, the size of the 2Sylow 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 2subgroups are generalized quaternion cannot be simple.
See also
Notes
 See also a table from Wolfram Alpha
 See Hall (1999), p. 190
 See Kurosh (1979), p. 67
 Johnson 1980, pp. 44–45
 Artin 1991
 Dean, Richard (1981). "A Rational Polynomial whose Group is the Quaternions". The American Mathematical Monthly. 88 (1): 42–45. JSTOR 2320711.
 Roman, Steven (2011). Fundamentals of Group Theory: An Advanced Approach. Springer. pp. 347–348. ISBN 9780817683016.
 Some authors (e.g., Rotman 1995, 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.
 Brown 1982, p. 98
 Brown 1982, p. 101, exercise 1
 Cartan & Eilenberg 1999, Theorem 11.6, p. 262
 Brown 1982, Theorem 4.3, p. 99
References
 Artin, Michael (1991), Algebra, Prentice Hall, ISBN 9780130047632
 Brown, Kenneth S. (1982), Cohomology of groups (3 ed.), SpringerVerlag, ISBN 9780387906881
 Cartan, Henri; Eilenberg, Samuel (1999), Homological Algebra, Princeton University Press, ISBN 9780691049915
 Coxeter, H. S. M. and Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: SpringerVerlag. ISBN 0387092129.CS1 maint: multiple names: authors list (link)
 Dean, Richard A. (1981) "A rational polynomial whose group is the quaternions", American Mathematical Monthly 88:42–5.
 Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 9780828403016, MR 0569209
 Johnson, David L. (1980), Topics in the theory of group presentations, Cambridge University Press, ISBN 9780521231084, MR 0695161
 Rotman, Joseph J. (1995), An introduction to the theory of groups (4 ed.), SpringerVerlag, ISBN 9780387942858
 P.R. Girard (1984) "The quaternion group and modern physics", European Journal of Physics 5:25–32.
 Hall, Marshall (1999), The theory of groups (2 ed.), AMS Bookstore, ISBN 0821819674
 Kurosh, Alexander G. (1979), Theory of Groups, AMS Bookstore, ISBN 0828401071