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| Quaternion group | |
|---|---|
| Name | Quaternion group |
| Notation | Q8 |
| Presentation | \langle i,j \mid i^4=1,\; i^2=j^2,\; j^{-1}ij=i^{-1}\rangle |
Quaternion group is a non-abelian finite group of order 8 arising in the study of division algebras, topology, and group theory. It is a central example in the classification of finite p-groups and appears in connections with the quaternion algebra, Clifford algebra, and symmetry groups of low-dimensional manifolds. The group is historically tied to work of William Rowan Hamilton and later algebraists who studied non-commutative structures and group cohomology.
The group is most commonly given by the presentation \langle i,j \mid i^4=1,\; i^2=j^2,\; j^{-1}ij=i^{-1}\rangle, where the element i has order 4 and j conjugates i to its inverse; this realization connects to the classical algebra of Hamiltonian quaternions discovered by William Rowan Hamilton. Alternative presentations include generators corresponding to the quaternion units ±1, ±i, ±j, ±k, each matching relations in the division ring of quaternions. The center is generated by −1, reflecting links to central extensions like those studied in work of Évariste Galois and later formalized by Camille Jordan.
Q8 is a non-abelian 2-group and a smallest example of a non-cyclic group with a unique element of order 2, placing it in discussions alongside the dihedral group D4 and the Klein four-group. Its exponent is 4, and it is Hamiltonian in the sense of being a non-abelian group whose every subgroup is normal, a property appearing in results associated with Philip Hall and Otto Hölder. The group fits into short exact sequences and central extensions studied by Emil Artin and appears in classification theorems for groups of small order used by G. A. Miller and Otto Schmidt.
Q8 has a center Z(Q8)=\{±1\} isomorphic to the cyclic group of order 2, and three maximal cyclic subgroups of order 4 generated by i, j, and k; these subgroups intersect pairwise in the center, a structure comparable to subgroup lattices considered by Marshall Hall Jr. and Bertram Huppert. Quotienting by the center yields a group of order 4 isomorphic to the Klein four-group, linking to classical results of Frobenius on factor groups. Normality of all subgroups makes Q8 a standard example in texts by Joseph Gallian and Daniel Gorenstein illustrating subgroup behavior in finite groups.
Linear representations of Q8 over the complex numbers decompose into irreducible representations including one-dimensional sign-type representations and a single two-dimensional irreducible representation, as treated in the representation theory developed by Issai Schur and expanded in works by Frobenius and I. M. Gelfand. Over fields of characteristic 2 the representation theory changes markedly, connecting to modular representation theory studied by Richard Brauer and J. A. Green. Q8-modules arise in classification problems for group algebras and in the study of projective representations and Schur multipliers, topics advanced by Issai Schur and later authors like Alperin.
Q8 appears as a subgroup of the unit group of Hamilton's quaternion algebra and in the automorphism groups of certain lattices studied by John Conway and J. H. Conway. It is central in examples in low-dimensional topology, for instance as the binary extension of rotational symmetry in SU(2) that double-covers SO(3), a relationship exploited in treatments by Henri Poincaré and in the theory of spin groups used in mathematical physics by Paul Dirac and Eugene Wigner. In crystallography and molecular symmetry texts by Linus Pauling and Frederick Seitz analogous double-cover phenomena are discussed. Q8 also serves as a counterexample in group-theoretic questions posed by Camille Jordan and appears in classification lists compiled by Bertrand Russell-era group theorists and modern compendia by Theodore Gamelin.
Generalizations include generalized quaternion groups Q_{2^n}, which extend Q8 to order 2^n and are important in the classification of finite p-groups pursued by Philip Hall and Gustav A. Hedlund. Related non-abelian groups of order 8 such as the dihedral group D4 and the semidihedral groups are contrasted in surveys by H. S. M. Coxeter and Donald J. Lewis. The role of Q8 in central extensions and in the theory of spin groups ties it to higher-dimensional Clifford algebra constructions used by Elie Cartan and Raoul Bott.
Category:Finite groups