Editing Quaternion group

{{Short description|Non-abelian group of order eight}}
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{{Group theory sidebar |Finite}}
[[Image:GroupDiagramQ8.svg|240px|thumb|[[Cycle graph (group)|Cycle diagram]] of Q<sub>8</sub>. 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 i<sup>2</sup> = {{overline|e}}, i<sup>3</sup> = {{overline|i}} and i<sup>4</sup> = e. The red cycle also reflects that {{overline|i}}<sup>2</sup> = {{overline|e}}, {{overline|i}}<sup>3</sup> = i and {{overline|i}}<sup>4</sup> = e.]]

In [[group theory]],  the '''quaternion group''' Q<sub>8</sub> (sometimes just denoted by Q) is a [[nonabelian group|non-abelian]] [[group (mathematics)|group]] of [[Group order|order]] eight, isomorphic to the eight-element subset
<math>\{1,i,j,k,-1,-i,-j,-k\}</math> of the [[quaternion]]s under multiplication. It is given by the [[presentation of a group|group presentation]]

:<math>\mathrm{Q}_8 = \langle \bar{e},i,j,k \mid \bar{e}^2 = e, \;i^2 = j^2 = k^2 = ijk = \bar{e} \rangle ,</math>

where ''e'' is the identity element and {{overline|''e''}} [[commutativity|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 Q<sub>8</sub> is
:<math>\mathrm{Q}_8 = \langle a,b \mid a^4 = e, a^2 = b^2, ba = a^{-1}b\rangle.</math>

Like many other finite groups, it [[Inverse Galois problem|can be realized]] as the [[#Galois group|Galois group]] of a certain field of [[algebraic number]]s.<ref name=":0" /> 

== Compared to dihedral group ==

The quaternion group Q<sub>8</sub> has the same order as the [[dihedral group]] [[Examples of groups#The symmetry group of a square: dihedral group of order 8|D<sub>4</sub>]], but a different structure, as shown by their Cayley and cycle graphs:

{| class="wikitable" style="width: 500px; text-align: center;"
!
! Q<sub>8</sub>
! [[dihedral group of order 8|D<sub>4</sub>]]
|-
! [[Cayley graph]]
|style="vertical-align:top;"| [[Image:Cayley graph Q8.svg|200px]]<br/>Red arrows connect ''g''<span style="color:red;">→</span>''gi'', green connect ''g''<span style="color:lightgreen;">→</span>''gj''.
|style="vertical-align:top;"| [[File:Dih 4 Cayley Graph; generators a, b.svg|220px]]
|-
! [[Cycle graph (algebra)|Cycle graph]]
| [[File:GroupDiagramQ8.svg|120px]]
| [[File:Dih4 cycle graph.svg|120px]]
|}

In the diagrams for D<sub>4</sub>, the group elements are marked with their action on a letter F in the defining representation '''R'''<sup>2</sup>. The same cannot be done for Q<sub>8</sub>, since it has no faithful representation in '''R'''<sup>2</sup> or '''R'''<sup>3</sup>. D<sub>4</sub> can be realized as a subset of the [[split-quaternion]]s in the same way that Q<sub>8</sub> can be viewed as a subset of the quaternions.

== Cayley table ==

The [[Cayley table]] (multiplication table) for Q<sub>8</sub> is given by:<ref>See also [http://www.wolframalpha.com/input/?i=Quaternion+group a table] from [[Wolfram Alpha]]</ref>

{|class="wikitable" style="text-align:right"
! × || e || {{overline|e}} || i || {{overline|i}} || j || {{overline|j}} || k || {{overline|k}}
|-
! e
| e || {{overline|e}} ||bgcolor=#ffcccc| i ||bgcolor=#ffcccc| {{overline|i}} ||bgcolor=#ccffcc| j ||bgcolor=#ccffcc| {{overline|j}} ||bgcolor=#ccccff| k ||bgcolor=#ccccff| {{overline|k}}
|-
! {{overline|e}}
| {{overline|e}} || e ||bgcolor=#ffcccc| {{overline|i}} ||bgcolor=#ffcccc| i ||bgcolor=#ccffcc| {{overline|j}} ||bgcolor=#ccffcc| j ||bgcolor=#ccccff| {{overline|k}} ||bgcolor=#ccccff| k 
|-
! i
|bgcolor=#ffcccc| i ||bgcolor=#ffcccc| {{overline|i}} || {{overline|e}} || e ||bgcolor=#ccccff| k ||bgcolor=#ccccff| {{overline|k}} ||bgcolor=#ccffcc| {{overline|j}} ||bgcolor=#ccffcc| j
|-
! {{overline|i}}
|bgcolor=#ffcccc| {{overline|i}} ||bgcolor=#ffcccc| i || e || {{overline|e}} ||bgcolor=#ccccff| {{overline|k}} ||bgcolor=#ccccff| k ||bgcolor=#ccffcc| j ||bgcolor=#ccffcc| {{overline|j}}
|-
! j
|bgcolor=#ccffcc| j ||bgcolor=#ccffcc| {{overline|j}} ||bgcolor=#ccccff| {{overline|k}} ||bgcolor=#ccccff| k || {{overline|e}} || e ||bgcolor=#ffcccc| i ||bgcolor=#ffcccc| {{overline|i}}
|-
! {{overline|j}}
|bgcolor=#ccffcc| {{overline|j}} ||bgcolor=#ccffcc| j ||bgcolor=#ccccff| k ||bgcolor=#ccccff| {{overline|k}} || e || {{overline|e}} ||bgcolor=#ffcccc| {{overline|i}} ||bgcolor=#ffcccc| i
|-
! k
|bgcolor=#ccccff| k ||bgcolor=#ccccff| {{overline|k}} ||bgcolor=#ccffcc| j ||bgcolor=#ccffcc| {{overline|j}} ||bgcolor=#ffcccc| {{overline|i}} ||bgcolor=#ffcccc| i || {{overline|e}} || e
|-
! {{overline|k}}
|bgcolor=#ccccff| {{overline|k}} ||bgcolor=#ccccff| k ||bgcolor=#ccffcc| {{overline|j}} ||bgcolor=#ccffcc| j ||bgcolor=#ffcccc| i ||bgcolor=#ffcccc| {{overline|i}} || e || {{overline|e}}
|}

== Properties ==

The elements ''i'', ''j'', and ''k'' all have [[order (group theory)|order]] four in Q<sub>8</sub> and any two of them generate the entire group. Another [[presentation of a group|presentation]] of Q<sub>8</sub><ref name="Johnson44-45">{{harvnb|Johnson|1980|loc=pp. 44&ndash;45}}</ref> based in only two elements to skip this redundancy is:

:<math>\left \langle x,y \mid x^4 = 1, x^2 = y^2, y^{-1}xy = x^{-1} \right \rangle.</math>

For instance, writing the group elements in [[Monomial order#lexicographic order|lexicographically]] minimal normal forms, one may identify: <blockquote><math>\{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 \}. </math>  </blockquote>The quaternion group has the unusual property of being [[Hamiltonian group|Hamiltonian]]: Q<sub>8</sub> is non-abelian, but every [[subgroup]] is [[normal subgroup|normal]].<ref>See Hall (1999), [https://books.google.com/books?id=oyxnWF9ssI8C&pg=PA190 p. 190]</ref> Every Hamiltonian group contains a copy of Q<sub>8</sub>.<ref>See Kurosh (1979), [https://books.google.com/books?id=rp9c0nyjkbgC&pg=PA67 p. 67]</ref>

The quaternion group Q<sub>8</sub> and the dihedral group D<sub>4</sub> are the two smallest examples of a [[nilpotent group|nilpotent]] non-abelian group.

The [[center of a group|center]] and the [[commutator subgroup]] of Q<sub>8</sub> is the subgroup <math>\{e,\bar{e}\}</math>. The [[inner automorphism group]] of Q<sub>8</sub> is given by the group modulo its center, i.e. the [[factor group]] <math>\mathrm{Q}_8/\{e,\bar{e}\},</math> which is [[isomorphic]] to the [[Klein four-group]] V. The full [[automorphism group]] of Q<sub>8</sub> is [[isomorphic]] to S<sub>4</sub>, the [[symmetric group]] on four letters (see ''Matrix representations'' below), and the [[outer automorphism group]] of Q<sub>8</sub> is thus S<sub>4</sub>/V, which is isomorphic to S<sub>3</sub>.

The quaternion group Q<sub>8</sub> has five conjugacy classes, <math>\{e\}, \{\bar{e}\}, \{i,\bar{i}\}, \{j,\bar{j}\}, \{k,\bar{k}\},</math> and so five [[irreducible representation]]s over the complex numbers, with dimensions 1, 1, 1, 1, 2:

'''Trivial representation'''.

'''Sign representations with i, j, k-kernel''': Q<sub>8</sub> has three maximal normal subgroups: the cyclic subgroups generated by i, j, and k respectively. For each maximal normal subgroup ''N'', we obtain a one-dimensional representation factoring through the 2-element [[quotient group]] ''G''/''N''. The representation sends elements of ''N'' to 1, and elements outside ''N'' to −1.

'''2-dimensional representation''': Described below in ''Matrix representations''. It is not [[Real representation|realizable over the real numbers]], but is a complex representation: indeed, it is just the quaternions <math>\mathbb{H}</math> considered as an algebra over <math>\mathbb C</math>, and the action is that of left multiplication by <math>Q_8\subset \mathbb H </math>.

The [[character table]] of Q<sub>8</sub> turns out to be the same as that of D<sub>4</sub>: 
{| class="wikitable"
|-
! Representation(ρ)/Conjugacy class !! { e } !! { {{overline|e}} } !! { i, {{overline|i}} } !! { j, {{overline|j}} } !! { k, {{overline|k}} }
|-
| Trivial representation || 1 || 1 || 1 || 1 || 1
|-
| Sign representation with i-kernel || 1 || 1 || 1 || −1 || −1
|-
| Sign representation with j-kernel || 1 || 1 || −1 || 1 || −1
|-
| Sign representation with k-kernel || 1 || 1 || −1 || −1 || 1
|-
| 2-dimensional representation || 2 || −2 || 0 || 0 || 0
|}

Nevertheless, all the irreducible characters <math>\chi_\rho</math> in the rows above have real values, this gives the [[Semisimple algebra#Classification|decomposition]] of the real [[group ring|group algebra]] of <math>G = \mathrm{Q}_8</math> into minimal two-sided [[Ideal (ring theory)|ideals]]:

:<math>\R[\mathrm{Q}_8]=\bigoplus_\rho (e_\rho),</math>

where the [[Idempotent (ring theory)|idempotents]] <math>e_\rho\in \R[\mathrm{Q}_8]</math> correspond to the irreducibles:

:<math>e_\rho = \frac{\dim(\rho)}{|G|}\sum_{g\in G} \chi_\rho(g^{-1})g,</math>

so that

:<math>\begin{align}
e_{\text{triv}} &= \tfrac 18(e + \bar e + i +\bar i+j+\bar j+k+\bar k) \\
e_{i\text{-ker}} &= \tfrac 18(e + \bar e + i +\bar i-j-\bar j-k-\bar k) \\
e_{j\text{-ker}} &= \tfrac 18(e + \bar e - i -\bar i+j+\bar j-k-\bar k) \\
e_{k\text{-ker}} &= \tfrac 18(e + \bar e - i -\bar i-j-\bar j+k+\bar k) \\
e_{2} &= \tfrac 28(2e - 2\bar e) = \tfrac 12(e - \bar e)
\end{align}</math>

Each of these irreducible ideals is isomorphic to a real [[central simple algebra]], the first four to the real field <math>\R</math>. The last ideal <math>(e_2)</math> is isomorphic to the [[skew field]] of [[quaternion]]s <math>\mathbb{H}</math> by the correspondence:

:<math>\begin{align}
\tfrac12(e-\bar e) &\longleftrightarrow 1, \\
\tfrac12(i-\bar i) &\longleftrightarrow i, \\
\tfrac12(j-\bar j) &\longleftrightarrow j, \\ 
\tfrac12(k-\bar k) &\longleftrightarrow k.
\end{align}</math>

Furthermore, the projection homomorphism <math>\R[\mathrm{Q}_8]\to (e_2)\cong \mathbb{H}</math> given by <math>r\mapsto re_2</math> has kernel ideal generated by the idempotent:

:<math>e_2^\perp = e_1+e_{i\text{-ker}}+e_{j\text{-ker}}+e_{k\text{-ker}} = 
\tfrac{1}{2}(e+\bar e), </math>

so the quaternions can also be obtained as the [[quotient ring]] <math>\R[\mathrm{Q}_8]/(e+\bar e)\cong \mathbb H</math>. Note that this is irreducible as a real representation of <math>Q_8</math>, but splits into two copies of the two-dimensional irreducible when extended to the complex numbers. Indeed, the complex group algebra is <math>\C[\mathrm{Q}_8] \cong \C^{\oplus 4} \oplus M_2(\C),</math> where <math>M_2(\C) \cong \mathbb{H} \otimes_{\R} \C</math> is the algebra of [[biquaternion]]s.

== Matrix representations ==
[[File:Quaternion group; Cayley table; subgroup of SL(2,C).svg|thumb|Multiplication table of quaternion group as a subgroup of [[Special linear group|SL]](2,[[Complex number|'''C''']]). The entries are represented by sectors corresponding to their arguments: 1 (green), ''i'' (blue), −1 (red), −''i'' (yellow).]]
The two-dimensional irreducible complex [[group representation|representation]] described above gives the quaternion group Q<sub>8</sub> as a subgroup of the [[general linear group]] <math>\operatorname{GL}(2, \C)</math>. The quaternion group is a multiplicative subgroup of the quaternion algebra:

:<math>\H = \R 1 + \R i + \R j + \R k= \C 1+ \C j,</math>

which has a [[regular representation]] <math>\rho:\H \to \operatorname{M}(2, \C)</math> by left multiplication on itself considered as a complex vector space with basis <math>\{1,j\},</math> so that <math>z \in \H</math> corresponds to the <math>\C</math>-linear mapping <math>\rho_z: a + jb \mapsto z\cdot(a + jb).</math> The resulting representation

:<math>\begin{cases} \rho:\mathrm{Q}_8 \to \operatorname{GL}(2,\C)\\ g\longmapsto\rho_g \end{cases}</math>

is given by:

:<math>\begin{matrix}
e \mapsto \begin{pmatrix}
         1 &          0 \\
         0 &          1
\end{pmatrix} &
i \mapsto \begin{pmatrix}
         i &          0 \\
         0 & \!\!\!\!-i
\end{pmatrix}&
j \mapsto \begin{pmatrix}
         0 & \!\!\!\!-1 \\
         1 &          0
\end{pmatrix}&
k \mapsto \begin{pmatrix}
         0 & \!\!\!\!-i \\
  \!\!\!-i &          0
\end{pmatrix} \\
\overline{e} \mapsto \begin{pmatrix}
  \!\!\!-1 &          0 \\
         0 & \!\!\!\!-1
\end{pmatrix} &
\overline{i} \mapsto \begin{pmatrix}
  \!\!\!-i &          0 \\
         0 &          i
\end{pmatrix}&
\overline{j} \mapsto \begin{pmatrix}
         0 &          1 \\
  \!\!\!-1 &          0
\end{pmatrix}&
\overline{k} \mapsto \begin{pmatrix}
         0 &          i \\
         i &          0
\end{pmatrix}.
\end{matrix}
</math>

Since all of the above matrices have unit determinant, this is a representation of Q<sub>8</sub> in the [[special linear group]] <math>\operatorname{SL}(2,\C)</math>.<ref>{{harvnb|Artin|1991}}</ref>

A variant gives a representation by [[Unitary matrix|unitary matrices]] (table at right). Let <math>g\in \mathrm{Q}_8</math> correspond to the linear mapping <math>\rho_g:a+bj\mapsto (a + bj)\cdot jg^{-1}j^{-1},</math> so that <math>\rho:\mathrm{Q}_8 \to \operatorname{SU}(2)</math> is given by:

:<math>\begin{matrix}
e \mapsto \begin{pmatrix}
         1 &          0 \\
         0 &          1
\end{pmatrix} &
i \mapsto \begin{pmatrix}
         i &          0 \\
         0 & \!\!\!\!-i
\end{pmatrix}&
j \mapsto \begin{pmatrix}
         0 &          1 \\
  \!\!\!-1 &          0
\end{pmatrix}&
k \mapsto \begin{pmatrix}
         0 &          i \\
         i &          0
\end{pmatrix} \\
\overline{e} \mapsto \begin{pmatrix}
  \!\!\!-1 &          0 \\
         0 & \!\!\!\!-1
\end{pmatrix} &
\overline{i} \mapsto \begin{pmatrix}
  \!\!\!-i &          0 \\
         0 &          i
\end{pmatrix}&
\overline{j} \mapsto \begin{pmatrix}
         0 & \!\!\!\!-1 \\
         1 &          0
\end{pmatrix}&
\overline{k} \mapsto \begin{pmatrix}
         0 & \!\!\!\!-i \\
  \!\!\!-i &          0
\end{pmatrix}.
\end{matrix}</math>

It is worth noting that physicists exclusively use a different convention for the <math>\operatorname{SU}(2)</math> matrix representation to make contact with the usual [[Spin (physics)#Pauli matrices|Pauli matrices]]:

:<math>\begin{matrix}
&e \mapsto \begin{pmatrix}
         1 &          0 \\
         0 &          1
\end{pmatrix} = \quad\, 1_{2\times2} 
&i \mapsto \begin{pmatrix}
         0 &          \!\!\!-i\! \\
         \!\!-i\!\! &          0
\end{pmatrix} = -i \sigma_x
&j \mapsto \begin{pmatrix}
         0 &          \!\!\!-1\! \\
         1 &           0
\end{pmatrix} = -i \sigma_y
&k \mapsto \begin{pmatrix}
         \!\!-i\!\! &          0 \\
          0 &          i
\end{pmatrix} = -i \sigma_z\\
&\overline{e} \mapsto \begin{pmatrix}
  \!\!-1\! &          0 \\
         0 & \!\!\!-1\!
\end{pmatrix} = -1_{2\times2}
&\overline{i} \mapsto \begin{pmatrix}
         0 &          i \\
         i &          0
\end{pmatrix} = \,\,\,\, i \sigma_x
&\overline{j} \mapsto \begin{pmatrix}
         0 &          1 \\
        \!\!-1\!\! &          0
\end{pmatrix} = \,\,\,\, i \sigma_y
&\overline{k} \mapsto \begin{pmatrix}
         i &           0 \\
         0 &          \!\!\!-i\!
\end{pmatrix} = \,\,\,\, i \sigma_z.
\end{matrix}</math>

This particular choice is convenient and elegant when one describes [[Spinor|spin-1/2 states]] in the <math>(\vec{J}^2, J_z)</math> basis and considers [[Angular momentum operator#Derivation using ladder operators|angular momentum ladder operators]] <math>J_{\pm} = J_x \pm iJ_y.</math>

[[File:Quaternion group; Cayley table; subgroup of SL(2,3).svg|thumb|Multiplication table of the quaternion group as a subgroup of [[:File:SL(2,3); Cayley table.svg|SL(2,3)]]. The field elements are denoted 0, +, −.]]

There is also an important action of Q<sub>8</sub> on the 2-dimensional vector space over the [[finite field]] <math>\mathbb{F}_3 =\{0, 1, -1\}</math> (table at right). A [[Modular representation theory|modular representation]] <math>\rho: \mathrm{Q}_8 \to \operatorname{SL}(2, 3)</math> is given by

:<math>\begin{matrix}
e \mapsto \begin{pmatrix}
         1 &          0 \\
         0 &          1
\end{pmatrix} &
i \mapsto \begin{pmatrix}
         1 &          1 \\
         1 & \!\!\!\!-1
\end{pmatrix} &
j \mapsto \begin{pmatrix}
  \!\!\!-1 &          1 \\
         1 &          1
\end{pmatrix} &
k \mapsto \begin{pmatrix}
         0 & \!\!\!\!-1 \\
         1 &          0
\end{pmatrix} \\
\overline{e} \mapsto \begin{pmatrix}
  \!\!\!-1 &          0 \\
         0 & \!\!\!\!-1
\end{pmatrix} &
\overline{i} \mapsto \begin{pmatrix}
  \!\!\!-1 & \!\!\!\!-1 \\
  \!\!\!-1 &          1
\end{pmatrix} &
\overline{j} \mapsto \begin{pmatrix}
         1 & \!\!\!\!-1 \\
  \!\!\!-1 & \!\!\!\!-1
\end{pmatrix} &
\overline{k} \mapsto \begin{pmatrix}
         0 &          1 \\
  \!\!\!-1 &          0
\end{pmatrix}.
\end{matrix}</math>

This representation can be obtained from the [[Field extension|extension field]]:

:<math> \mathbb{F}_9 = \mathbb{F}_3 [k] = \mathbb{F}_3 1 + \mathbb{F}_3 k,</math>

where <math>k^2=-1</math> and the multiplicative group <math>\mathbb{F}_9^{\times}</math> has four generators, <math>\pm(k\pm1),</math> of order 8. For each <math>z \in \mathbb{F}_9,</math> the two-dimensional <math>\mathbb{F}_3</math>-vector space <math>\mathbb{F}_9</math> admits a linear mapping:

:<math>\begin{cases} \mu_z: \mathbb{F}_9 \to \mathbb{F}_9 \\ \mu_z(a+bk)=z\cdot(a+bk) \end{cases}</math>

In addition we have the [[Frobenius endomorphism|Frobenius automorphism]] <math>\phi(a+bk)=(a+bk)^3</math> satisfying <math>\phi^2 = \mu_1 </math> and <math>\phi\mu_z = \mu_{\phi(z)}\phi.</math> Then the above representation matrices are:

:<math>\begin{align}
\rho(\bar e) &=\mu_{-1}, \\
\rho(i) &=\mu_{k+1}\phi, \\
\rho(j)&=\mu_{k-1} \phi, \\
\rho(k)&=\mu_{k}.
\end{align}</math>

This representation realizes Q<sub>8</sub> as a [[normal subgroup]] of {{nowrap|GL(2, 3)}}. Thus, for each matrix <math>m\in \operatorname{GL}(2,3)</math>, we have a group automorphism

:<math>\begin{cases} \psi_m:\mathrm{Q}_8\to\mathrm{Q}_8 \\ \psi_m(g)=mgm^{-1} \end{cases}</math>

with <math>\psi_I =\psi_{-I}=\mathrm{id}_{\mathrm{Q}_8}.</math> In fact, these give the full automorphism group as:

:<math>\operatorname{Aut}(\mathrm{Q}_8) \cong \operatorname{PGL}(2, 3) = \operatorname{GL}(2,3)/\{\pm I\}\cong S_4.</math>

This is isomorphic to the symmetric group S<sub>4</sub> since the linear mappings <math>m:\mathbb{F}_3^2 \to \mathbb{F}_3^2</math> permute the four one-dimensional subspaces of <math>\mathbb{F}_3^2,</math> i.e., the four points of the [[projective space]] <math>\mathbb{P}^1 (\mathbb{F}_3) = \operatorname{PG}(1,3).</math>

Also, this representation permutes the eight non-zero vectors of <math>\mathbb{F}_3^2,</math> giving an embedding of Q<sub>8</sub> in the [[symmetric group]] S<sub>8</sub>, in addition to the embeddings given by the regular representations.

==Galois group==
[[Richard Dedekind]] considered the field <math>\mathbb{Q}[\sqrt{2}, \sqrt{3}]</math> in attempting to relate the quaternion group to [[Galois theory]].<ref>[[Richard Dedekind]] (1887) "Konstrucktion der Quaternionkörpern", Ges. math. Werk II 376–84</ref> In 1936 [[Ernst Witt]] published his approach to the quaternion group through Galois theory.<ref>[[Ernst Witt]] (1936) "Konstruktion von galoisschen Körpern..."[[Crelle's Journal]] 174: 237-45</ref>

In 1981, Richard Dean showed the quaternion group can be realized as the [[Galois group]] Gal(T/'''Q''') where '''Q''' is the field of [[rational number]]s and T is the [[splitting field]] of the polynomial
:<math>x^8 - 72 x^6 + 180 x^4 - 144 x^2 + 36</math>.
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.<ref name=":0">{{cite journal
 | last = Dean | first =  Richard
 | year = 1981
 | title = A Rational Polynomial whose Group is the Quaternions
 | journal = [[American Mathematical Monthly|The American Mathematical Monthly]]
 | volume = 88 | issue = 1 | pages = 42–45
 | doi = 10.2307/2320711
 | jstor = 2320711
}}</ref>

==Generalized quaternion group==
A '''generalized quaternion group''' Q<sub>4''n''</sub> of order 4''n'' is defined by the presentation<ref name="Johnson44-45" />
:<math>\langle x,y \mid x^{2n} = y^4 = 1, x^n = y^2, y^{-1}xy = x^{-1}\rangle</math>
for an integer {{nowrap|''n'' ≥ 2}}, with the usual quaternion group given by ''n'' = 2.<ref>Some authors (e.g., {{harvnb|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.</ref> [[Harold Scott MacDonald Coxeter|Coxeter]] calls Q<sub>4''n''</sub> the [[dicyclic group]] <math>\langle 2, 2, n\rangle</math>, a special case of the [[binary polyhedral group]] <math>\langle \ell, m, n\rangle</math> and related to the [[polyhedral group]] <math>(p,q,r)</math> and the [[dihedral group]] <math>(2,2,n)</math>. The generalized quaternion group can be realized as the subgroup of <math>\operatorname{GL}_2(\Complex)</math> generated by

:<math>\left(\begin{array}{cc}
               \omega_n & 0 \\
               0 & \overline{\omega}_n
             \end{array}
          \right)
        \mbox{ and }
        \left(\begin{array}{cc}
                0 & -1 \\
                1 & 0
              \end{array}
          \right)
</math>

where <math>\omega_n = e^{i\pi/n}</math>.<ref name="Johnson44-45" /> It can also be realized as the subgroup of unit quaternions generated by<ref>{{harvnb|Brown|1982}}, p. 98</ref> <math>x=e^{i\pi/n}</math> and <math>y=j</math>.

The generalized quaternion groups have the property that every [[abelian group|abelian]] subgroup is cyclic.<ref>{{harvnb|Brown|1982}}, p. 101, exercise 1</ref> It can be shown that a finite [[p-group|''p''-group]] with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above.<ref>{{harvnb|Cartan|Eilenberg|1999}}, Theorem 11.6, p. 262</ref> 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.<ref>{{harvnb|Brown|1982}}, Theorem 4.3, p. 99</ref> In particular, for a finite field ''F'' with odd characteristic, the 2-Sylow subgroup of SL<sub>2</sub>(''F'') is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group, {{harv|Gorenstein|1980|p=42}}.  Letting ''p<sup>r</sup>'' be the size of ''F'', where ''p'' is prime, the size of the 2-Sylow subgroup of SL<sub>2</sub>(''F'') is 2<sup>''n''</sup>, where {{nowrap|1=''n'' = ord<sub>2</sub>(''p''<sup>2</sup> − 1) + ord<sub>2</sub>(''r'')}}.

The [[Brauer–Suzuki theorem]] shows that the groups whose Sylow 2-subgroups are generalized quaternion cannot be simple.

Another terminology reserves the name "generalized quaternion group" for a dicyclic group of order a power of 2,<ref name="Roman">{{cite book|last1=Roman|first1=Steven|title=Fundamentals of Group Theory: An Advanced Approach|publisher=Springer|year=2011|isbn=9780817683016|pages=347–348|author-link1=Steven Roman}}</ref> which admits the presentation
:<math>\langle x,y \mid x^{2^m} = y^4 = 1, x^{2^{m-1}} = y^2, y^{-1}xy = x^{-1}\rangle.</math>
<!-- more ref,  http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/genquat.pdf 
http://groupprops.subwiki.org/wiki/Generalized_quaternion_group
http://enc.slider.com/Enc/Quaternion_group
-->==See also==
*[[16-cell]]
*[[Binary tetrahedral group]]
*[[Clifford algebra]]
*[[Dicyclic group]]
*[[Hurwitz integral quaternion]]
*[[List of small groups]]

==Notes==
{{Reflist}}

==References==
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==External links==
* {{MathWorld | urlname = QuaternionGroup | title = Quaternion group}}
* [http://groupnames.org/#?quaternion Quaternion groups on GroupNames]
* Quaternion group on [https://groupprops.subwiki.org/wiki/Quaternion_group GroupProps]
* Conrad, Keith. [https://kconrad.math.uconn.edu/blurbs/grouptheory/genquat.pdf "Generalized Quaternions"]

[[Category:Group theory]]
[[Category:Finite groups]]
[[Category:Quaternions|group]]