Editing Quaternion group (section)

== 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.