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quaternion group

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Idea

The quaternion group of order 8, Q 8Q_8, is the finite subgroup of SU(2) Q 8SU(2)S 3Q_8 \subset SU(2) \simeq S^3 \subset \mathbb{H} of unit quaternions which consists of the canonical four basis-quaternions and their negatives:

Q 8={±1,±i±j±k}. Q_8 \;=\; \big\{ \pm 1, \, \pm i \, \pm j \, \pm k \big\} \,.

This is isomorphic to the binary dihedral group of the same order Q 82D 4Q_8 \simeq 2 D_4. As such, the Dynkin diagram that corresponds to Q 8Q_8 under the ADE-classification of finite subgroups of SU(2) is D4, the triality-invariant one.

graphics grabbed from Wikipedia here

This order-8 quaternion group Q 8Q_8 is the first in a row of generalized quaternion groups, Q 2 nQ_{2^n}, which are also examples of dicyclic groups, which class forms part of an even larger family. We will treat both general dicyclic groups and the specific example of the quaternion group together.

Definitions

The dicyclic of order 4n4n, n2n\geq 2, is the group Dic nDic_n defined by the presentation x,y|x 2n=x ny 2=y 1xyx=1\langle x,y | x^{2n}= x^{n} y^{-2}=y^{-1}x y x=1\rangle.

The quaternion group (of order 8) is then Dic nDic_n for n=2n=2.

The generalised quaternion group of order 2 k+12^{k+1} is Dic nDic_n with n=2 k1n= 2^{k-1}.

Properties

General

  • Dic nDic_n is (isomorphic to) a finite subgroup of *\mathbb{H}^\ast as can be seen by taking generators x=jx=j and y=cos(π/n)+isin(π/n)y=\cos(\pi/n) + i\sin(\pi/n). For n=2n=2 this simply yields the subgroup generated by ii and jj.

  • Dic nDic_n has another presentation as R,S,T|R 2=S 2=T n=RST\langle R, S, T | R^2=S^2=T^n=R S T\rangle. RSTR S T as a power of each of the generators is central and Dic n/RST=D 2nDic_n/\langle R S T\rangle= D_{2n}, where D 2n=R,S,T|R 2=S 2=T n=RST=1D_{2n}=\langle R, S, T | R^2=S^2=T^n=R S T=1\rangle is the dihedral group of order 2n2n.

  • Q 8Q_8 is a Hamiltonian group i.e. a non-abelian group such that every subgroup is normal. Moreover, a general structure theorem for Hamiltonian groups by Baer (1933) says that every Hamiltonian group has a direct product group-decomposition containing Q 8Q_8 as a factor hence, in particular, every Hamiltonian group contains Q 8Q_8 as a subgroup! (cf. Scott (1987, p.253))

  • Q 8Q_8 is the multiplicative part of the quaternionic near-field J 9J_9. (cf. Weibel (2007))

Proposition

(inclusion of Q 8Q_8 into finite subgroups of SU(2))

Among the finite subgroups of SU(2) (hence among all “finite quaternion groups”) the quaternion group of order 8, Q 8Q_8 is a proper subgroup precisely of the three exceptional cases:

(e.g. Koca-Moc-Koca 16, p. 8, pointing to Coxeter-Moser 65 and Coxeter 73)

Character table

linear representation theory of binary dihedral group 2D 42 D_4

== dicyclic group Dic 2Dic_2 == quaternion group Q 8Q_8

\,

group order: |2D 4|=8{\vert 2D_4\vert} = 8

conjugacy classes:124A4B4C
their cardinality:11222

\,

splitting field(α,β)\mathbb{Q}(\alpha, \beta) with α 2+β 2=1\alpha^2 + \beta^2 = -1
field generated by characters\mathbb{Q}

character table over splitting field (α,β)\mathbb{Q}(\alpha,\beta)/complex numbers \mathbb{C}

irrep124A4B4CSchur index
ρ 1\rho_1111111
ρ 2\rho_211-11-11
ρ 3\rho_3111-1-11
ρ 4\rho_411-1-111
ρ 5\rho_52-20002

character table over rational numbers \mathbb{Q}/real numbers \mathbb{R}

irrep124A4B4C
ρ 1\rho_111111
ρ 2\rho_211-11-1
ρ 3\rho_3111-1-1
ρ 4\rho_411-1-11
ρ 5ρ 5\rho_5 \oplus \rho_54-4000

References

Matrix representation

There are lots of different ways of defining Q:=Q 8Q:=Q_8. One is that it is the subgroup of Gl(2,)Gl(2,\mathbb{C}) generated by the matrices

ξ=(i 0 0 i)\xi = \left(\array{i&0\\0&-i}\right)

and

η=(0 1 1 0).\eta =\left(\array{0&-1\\1&0}\right).

In this form it is a nice exercise to derive a presentation of Q 8Q_8. Clearly ξ 4=1\xi^4=1 and η\eta is not in ξ \langle \xi\rangle as is easiy checked, so the order of this group must be at least 8.

We note that η 2=ξ 2\eta^2 = \xi^2 and that ηξη 1=ξ 1\eta \xi \eta^{-1}= \xi^{-1}, so a guess for a presentation would be

x,y:x 4=1,y 2=x 2,yxy 1=x 1. \langle x,y : x^4=1, y^2=x^2, y x y^{-1}=x^{-1}\rangle.

Let us call GG the group presented by this presentation, then there is an obvious epimorphism from GG to QQ sending xx to ξ\xi and yy to η\eta. This is an isomorphism as will be clear if we show that the order of GG is less than of equal to 8. Now every element of GG can be written in the form x iy jx^i y^j with 0i30\leq i\leq 3 and 0j10\leq j\leq 1, since yx=x 1yy x=x^{-1}y so powers of yy can be shifted to the right in any expression and then if the resulting power of yy is greater than 2 we can use y 2=x 2y^2=x^2 to replace even powers of yy by powers of xx. We must therefore have that the group GG must contain at most 8 elements so the above presentation is a presentation of Q 8Q_8.

References

  • Kenneth S. Brown, Cohomology of Groups , GTM 87 Springer Heidelberg 1982. (pp.98-101)

  • H. S. M. Coxeter, The binary polyhedral groups, and other generalizations of the quaternion group , Duke Math. J. 7 no.1 (1940) pp.367–379.

  • T. Y. Lam, Hamilton’s Quaternions , pp.429-454 in Handbook of Algebra III , Elsevier Amsterdam 2004. (preprint)

  • W. R. Scott, Group Theory , Dover New York 1987. (pp.189-194, 252-254)

  • Charles Weibel, Survey of Non-Desarguesian Planes , Notices of the AMS 54 no.10 (2007) pp.1294–1303. (pdf)

  • Mehmet Koca, Ramazan Koç, Nazife Ozdes Koca, Two groups 2 3.PSL 2(7)2^3.PSL_2(7) and 2 3:PSL 2(7)2^3:PSL_2(7) of order 1344 (arXiv:1612.06107)

  • H.S.M. Coxeter, W. O. J. Moser, Generators and Relations for Discrete Groups, (Springer Verlag, 1965);

  • H.S.M. Coxeter, Regular Complex Polytopes (Cambridge; Cambridge University Press, 1973).

See also

Last revised on October 4, 2018 at 12:04:06. See the history of this page for a list of all contributions to it.