Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Quotient group (Semi-)direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable action Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
v t e

In group theory, the quaternion group Q₈ (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset ${\displaystyle \{1,i,j,k,-1,-i,-j,-k\}}$ of the quaternions under multiplication. It is given by the group presentation

${\displaystyle \mathrm {Q} _{8}=\langle {\bar {e}},i,j,k\mid {\bar {e}}^{2}=e,\;i^{2}=j^{2}=k^{2}=ijk={\bar {e}}\rangle ,}$

where e is the identity element and e commutes with the other elements of the group. These relations, discovered by W. R. Hamilton, also generate the quaternions as an algebra over the real numbers.

Another presentation of Q₈ is

${\displaystyle \mathrm {Q} _{8}=\langle a,b\mid a^{4}=e,a^{2}=b^{2},ba=a^{-1}b\rangle .}$

Like many other finite groups, it can be realized as the Galois group of a certain field of algebraic numbers.^[1]

Compared to dihedral group

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

	Q8	D4
Cayley graph	Red arrows connect g→gi, green connect g→gj.
Cycle graph

In the diagrams for D₄, the group elements are marked with their action on a letter F in the defining representation R². The same cannot be done for Q₈, since it has no faithful representation in R² or R³. D₄ can be realized as a subset of the split-quaternions in the same way that Q₈ can be viewed as a subset of the quaternions.

Cayley table

The Cayley table (multiplication table) for Q₈ is given by:^[2]

Properties

The elements i, j, and k all have order four in Q₈ and any two of them generate the entire group. Another presentation of Q₈^[3] based in only two elements to skip this redundancy is:

${\displaystyle \left\langle x,y\mid x^{4}=1,x^{2}=y^{2},y^{-1}xy=x^{-1}\right\rangle .}$

For instance, writing the group elements in lexicographically minimal normal forms, one may identify:

${\displaystyle \{e,{\bar {e}},i,{\bar {i}},j,{\bar {j}},k,{\bar {k}}\}\leftrightarrow \{e,x^{2},x,x^{3},y,x^{2}y,xy,x^{3}y\}.}$

The quaternion group has the unusual property of being Hamiltonian: Q₈ is non-abelian, but every subgroup is normal.^[4] Every Hamiltonian group contains a copy of Q₈.^[5]

The quaternion group Q₈ and the dihedral group D₄ are the two smallest examples of a nilpotent non-abelian group.

The center and the commutator subgroup of Q₈ is the subgroup ${\displaystyle \{e,{\bar {e}}\}}$. The inner automorphism group of Q₈ is given by the group modulo its center, i.e. the factor group ${\displaystyle \mathrm {Q} _{8}/\{e,{\bar {e}}\},}$ which is isomorphic to the Klein four-group V. The full automorphism group of Q₈ is isomorphic to S₄, the symmetric group on four letters (see Matrix representations below), and the outer automorphism group of Q₈ is thus S₄/V, which is isomorphic to S₃.

The quaternion group Q₈ has five conjugacy classes, ${\displaystyle \{e\},\{\stackrel{¯<!--\; ¯\; -->}{e}}$Zdroj:https://en.wikipedia.org?pojem=Quaternion_group
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.