By a finite rotation group one means a finite subgroup of a group of rotations, hence of a special orthogonal group or spin group or similar.
The finite subgroups of SO(3) and SU(2) follow an ADE classification (theorem below).
(ADE classification of finite subgroups of SO(3) and Spin(3) SU(2))
The finite subgroups of the special orthogonal group as well as the finite subgroups of the special unitary group SU(2) are, up to conjugation, given by the following classification:
ADE classification and McKay correspondence
Here under the double cover projection (using the exceptional isomorphism )
all the finite subgroups of except the odd-order cyclic groups are the preimages of the corresponding finite subgroups of , in that we have pullback diagrams
exhibiting the even order cyclic groups as subgroups of Spin(2), including the minimal case of the group of order 2
as well as
exhibiting the binary dihedral groups as sitting inside the Pin(2)-subgroup of Spin(3),
but only commuting diagrams
This goes back to (Klein 1884, chapter I). Full proof for is spelled out for instance in (Rees 05, theorem 11, De Visscher 11). The proof for the case of is spelled out in (Miller-Blichfeldt-Dickson 16) reviewed in (Serrano 14, section 2). The proof of the case for given the result for is spelled out in Keenan 03, theorem 4.
For classification of the finite subgroups of see (duVal 65, Conway-Smith 03)
For finite subgroups of Spin(4): MFF 12, appendix B.
In this classification, the symmetry group of the 120-cell and hence that of the 600-cell is the quotient group by the cyclic group of order 2 of the direct product group of two copies of the binary icosahedral group (SadocMosseri 89, p. 172, see MFF 12, table 16).
For classification of the finite subgroups of see Mecchia-Zimmermann 10
The subgroup lattice of SU(2) under the three exceptional finite subgroups 2T, 2O, 2I (from Theorem ) looks as follows:
This is obtained from the subgroup lattice as shown on GroupNames for SL(2,5) and CSU(2,3)
See also Goncalves-Guaschi 11, appendix.
(group cohomology of finite subgroups of SU(2))
Let be a finite subgroup of SU(2). Then its group cohomology with integer coefficients is as follows:
Here denotes the abelianization of and its cardinality, hence the cyclic group whose order is the cardinality of .
The group homology with integer coefficients is
For pointers to proofs see for instance Epa-Ganter 16, section 4.
In discussion of 11-dimensional supergravity on spacetimes with ADE-singularities, the special case
of Prop. , regarded as expressing orbifold cohomology of an ADE singularity, as shown under the brace, witnesses the possible torsion supergravity C-field flux of M5-branes wrapped on torsion homology 3-cycles (“discrete torsion”, see Aharony-Bergman-Jafferis 08, p. 8 and BDHKMMS 01, section 4.6.2).
See also at Platonic 2-group – Relation to String 2-group.
The classification in Theorem goes back to
Textbook accounts include
G. A. Miller, H. F. Blichfeldt, L. E. Dickson, Theory and applications of finite groups, Dover, New York, 1916
Klaus Lamotke, Regular Solids and Isolated Singularities, Vieweg 1986
Elmer Rees, Notes on Geometry, Springer 2005
see also
Complete proof of the classification of the finite subgroups of is also spelled out in
Based on the classification of the finite subgroups of , full proof of that of the finite subgroups of is spelled out in
See also
Discussion of the lattice of subgroups of the three exceptional subgroups is in
The universal higher central extension of finite subgroups of (“Platonic 2-groups”) are discussed in
Patrick du Val, Homographies, Quaternions and Rotations, Oxford Mathematical Monographs, Clarendon Press (1964)
also(?): Journal of the London Mathematical Society, Volume s1-40, Issue 1 (1965) (doi:10.1112/jlms/s1-40.1.569b)
John Conway, D. A. Smith, On quaternions and octonions: their geometry, arithmetic and
symmetry_ A K Peters Ltd., Natick, MA, 2003
Paul de Medeiros, José Figueroa-O'Farrill, appendix B of Half-BPS M2-brane orbifolds, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (arXiv:1007.4761, Euclid)
J. F. Sadoc, R. Mosseri, Icosahedral order, curved space and quasicrystals, in Jaric, Gratias (eds.) Extended icosahedral structures, 1989 (GoogleBooks)
Last revised on December 23, 2020 at 06:47:44. See the history of this page for a list of all contributions to it.