Jump to content

Quaternion group

From Wikipedia, the free encyclopedia

This is an old revision of this page, as edited by Maproom (talk | contribs) at 11:41, 28 November 2016 (Compared to dihedral group: changed prepositions). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element e = 1. For example, the cycle in red reflects the fact that i2 = e, i3 = i and i4 = e. The red cycle also reflects that i2 = e, i3 = i and i4 = e.

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 e is the identity element and e commutes with the other elements of the group.

Compared to dihedral group

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

Q8 Dih4
Cayley graph
Red arrows represent multiplication by i, green arrows by j.
Cycle graph

The dihedral group D4 arises in the split-quaternions in the same way that Q8 lies in the quaternions.

Cayley table

The Cayley table (multiplication table) for Q 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 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 as the quotient of the group ring R[Q] by the ideal defined by spanR({e+e, i+i, j+j, k+k}).[citation needed] 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 = xy.

The center and the commutator subgroup of Q is the subgroup {e,e}. The factor group Q/{e,e} 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 the symmetric group of degree 4, 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[7] when its order is a power of 2 and it is a dicyclic group.

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 Q4n 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 GL2(C) generated by

where ωn = e/n.[4] It can also be realized as the subgroup of unit quaternions generated by[9] x = e/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 p-group 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 p-group in which there is a unique subgroup of order p is either cyclic or a 2-group isomorphic to generalized quaternion group.[12] 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, (Gorenstein 1980, p. 42). 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. ^ a b c Johnson 1980, pp. 44–45
  5. ^ Artin 1991
  6. ^ Dean, Richard (1981). "A Rational Polynomial whose Group is the Quaternions". The American Mathematical Monthly. 88 (1): 42–45. JSTOR 2320711.
  7. ^ Roman, Steven (2011). Fundamentals of Group Theory: An Advanced Approach. Springer. pp. 347–348. ISBN 9780817683016.
  8. ^ 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.
  9. ^ Brown 1982, p. 98
  10. ^ Brown 1982, p. 101, exercise 1
  11. ^ Cartan & Eilenberg 1999, Theorem 11.6, p. 262
  12. ^ Brown 1982, Theorem 4.3, p. 99

References