Quaternion group

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Cycle diagram of Q. 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 i2 = e, i3 = i and i4 = e. The red cycle also reflects that i2 = e, i3 = i and i4 = e.

In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication. It is often denoted by Q or Q8, and is given by the group presentation

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

where e is the identity element and e commutes with the other elements of the group.

Compared to dihedral group

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

Q8 Dih4
Cayley graph
Red arrows represent multiplication by i, green arrows by j.
Cycle graph

The dihedral group D4 arises in the split-quaternions in the same way that Q8 lies in the quaternions.

Cayley table

The Cayley table (multiplication table) for Q is given by:[1]

× e e i i j j k k
e e e i i j j k k
e e e i i j j k k
i i i e e k k j j
i i i e e k k j j
j j j k k e e i i
j j j k k e e i i
k k k j j i i e e
k k k j j i i e e

Properties

The quaternion group has the unusual property of being Hamiltonian: every subgroup of Q is a normal subgroup, but the group is non-abelian.[2] Every Hamiltonian group contains a copy of Q.[3]

The quaternion group is the smallest example of nilpotent non-abelian group. Another example of nilpotent non-abelian group is dihedral group of order eight (generated by an element of order four and an element of order two that conjugates the element of order four to its inverse).

The quaternion group has five irreducible representations, and their dimensions are 1,1,1,1,2, respectively. The proof for this property is not hard, since the number of irreducible characters of the quaternion group is equal to the number of conjugacy classes of the quaternion group, which is five ( { e }, { e }, { i, i }, { j, j }, { k, k } ).

These five representations are:

Trivial representation

Sign representations with i,j,k-kernel: the quaternion group has three maximal normal subgroups: the cyclic subgroups generated by respectively. For each maximal normal subgroup, we obtain a one-dimensional representation with that subgroup as kernel. The representation sends elements inside the subgroup to 1, and elements outside the subgroup to -1.

2-dimensional representation: A representation :${\displaystyle \mathrm {Q} =\{\mathrm {e} ,{\bar {\mathrm {e} }},\mathrm {i} ,{\bar {\mathrm {i} }},\mathrm {j} ,{\bar {\mathrm {j} }},\mathrm {k} ,{\bar {\mathrm {k} }}\}\to \mathrm {GL} _{2}(\mathbf {C} )}$ stated in Matrix representations.

So the character table of the quaternion group is (actually it has the same character table as dihedral group D8):

Representation/Conjugacy class { e } { e } { i, i } { j, j } { k, k }
Trivial representation 1 1 1 1 1
Sign representations with i-kernel 1 1 1 -1 -1
Sign representations with j-kernel 1 1 -1 1 -1
Sign representations with k-kernel 1 1 -1 -1 1
2-dimensional representation 2 -2 0 0 0

In abstract algebra, one can construct a real four-dimensional vector space as the quotient of the group ring R[Q] by the ideal defined by spanR({e+e, i+i, j+j, k+k}).[citation needed] The result is a skew field called the quaternions. Note that this is not quite the same as the group algebra on Q (which would be eight-dimensional). Conversely, one can start with the quaternions and define the quaternion group as the multiplicative subgroup consisting of the eight elements {1, −1, i, −i, j, −j, k, −k}. The complex four-dimensional vector space on the same basis is called the algebra of biquaternions.

Note that i, j, and k all have order four in Q and any two of them generate the entire group. Another presentation of Q[4] demonstrating this is:

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

One may take, for instance, i = x, j = y and k = xy.

The center and the commutator subgroup of Q is the subgroup {e,e}. The factor group Q/{e,e} is isomorphic to the Klein four-group V. The inner automorphism group of Q is isomorphic to Q modulo its center, and is therefore also isomorphic to the Klein four-group. The full automorphism group of Q is isomorphic to the symmetric group of degree 4, S4, the symmetric group on four letters. The outer automorphism group of Q is then S4/V which is isomorphic to S3.

Matrix representations

Q. g. as a subgroup of SL(2,C)

The quaternion group can be represented as a subgroup of the general linear group GL2(C). A representation

${\displaystyle \mathrm {Q} =\{\mathrm {e} ,{\bar {\mathrm {e} }},\mathrm {i} ,{\bar {\mathrm {i} }},\mathrm {j} ,{\bar {\mathrm {j} }},\mathrm {k} ,{\bar {\mathrm {k} }}\}\to \mathrm {GL} _{2}(\mathbf {C} )}$

is given by

${\displaystyle {\begin{matrix}\mathrm {e} \mapsto {\begin{pmatrix}1&0\\0&1\end{pmatrix}}&\mathrm {i} \mapsto {\begin{pmatrix}i&0\\0&-i\end{pmatrix}}&\mathrm {j} \mapsto {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}&\mathrm {k} \mapsto {\begin{pmatrix}0&i\\i&0\end{pmatrix}}\\{\overline {\mathrm {e} }}\mapsto {\begin{pmatrix}-1&0\\0&-1\end{pmatrix}}&{\overline {\mathrm {i} }}\mapsto {\begin{pmatrix}-i&0\\0&i\end{pmatrix}}&{\overline {\mathrm {j} }}\mapsto {\begin{pmatrix}0&-1\\1&0\end{pmatrix}}&{\overline {\mathrm {k} }}\mapsto {\begin{pmatrix}0&-i\\-i&0\end{pmatrix}}\end{matrix}}}$

Since all of the above matrices have unit determinant, this is a representation of Q in the special linear group SL2(C). The standard identities for quaternion multiplication can be verified using the usual laws of matrix multiplication in GL2(C).[5]

Q. g. as a subgroup of SL(2,3)

There is also an important action of Q on the eight nonzero elements of the 2-dimensional vector space over the finite field F3. A representation

${\displaystyle \mathrm {Q} =\{\mathrm {e} ,{\bar {\mathrm {e} }},\mathrm {i} ,{\bar {\mathrm {i} }},\mathrm {j} ,{\bar {\mathrm {k} }},{\bar {\mathrm {k} }}\}\to \mathrm {GL} (2,3)}$

is given by

${\displaystyle {\begin{matrix}\mathrm {e} \mapsto {\begin{pmatrix}1&0\\0&1\end{pmatrix}}&\mathrm {i} \mapsto {\begin{pmatrix}1&1\\1&-1\end{pmatrix}}&\mathrm {j} \mapsto {\begin{pmatrix}-1&1\\1&1\end{pmatrix}}&\mathrm {k} \mapsto {\begin{pmatrix}0&-1\\1&0\end{pmatrix}}\\{\overline {\mathrm {e} }}\mapsto {\begin{pmatrix}-1&0\\0&-1\end{pmatrix}}&{\overline {\mathrm {i} }}\mapsto {\begin{pmatrix}-1&-1\\-1&1\end{pmatrix}}&{\overline {\mathrm {j} }}\mapsto {\begin{pmatrix}1&-1\\-1&-1\end{pmatrix}}&{\overline {\mathrm {k} }}\mapsto {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}\end{matrix}}}$

where {−1, 0, 1} are the three elements of F3. Since all of the above matrices have unit determinant over F3, this is a representation of Q in the special linear group SL(2, 3). Indeed, the group SL(2, 3) has order 24, and Q is a normal subgroup of SL(2, 3) of index 3.

Galois group

As Richard Dean showed in 1981, the quaternion group can be presented as the Galois group Gal(T/Q) where Q is the field of rational numbers and T is the splitting field, over Q, of the polynomial

${\displaystyle x^{8}-72x^{6}+180x^{4}-144x^{2}+36}$.

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.[6]

Generalized quaternion group

A group is called a generalized quaternion group[7] when its order is a power of 2 and it is a dicyclic group.

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

It is a part of more general class of dicyclic groups.

Some authors define [4] generalized quaternion group to be the same as dicyclic group.

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

for some integer n ≥ 2. This group is denoted Q4n and has order 4n.[8] Coxeter labels these dicyclic groups <2,2,n>, being a special case of the binary polyhedral group <l,m,n> and related to the polyhedral groups (p,q,r), and dihedral group (2,2,n). The usual quaternion group corresponds to the case n = 2. The generalized quaternion group can be realized as the subgroup of GL2(C) generated by

${\displaystyle \left({\begin{array}{cc}\omega _{n}&0\\0&{\overline {\omega }}_{n}\end{array}}\right){\mbox{ and }}\left({\begin{array}{cc}0&-1\\1&0\end{array}}\right)}$

where ωn = e/n.[4] It can also be realized as the subgroup of unit quaternions generated by[9] x = e/n and y = j.

The generalized quaternion groups have the property that every abelian subgroup is cyclic.[10] It can be shown that a finite p-group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above.[11] 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.[12] In particular, for a finite field F with odd characteristic, the 2-Sylow subgroup of SL2(F) is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group, (Gorenstein 1980, p. 42). Letting pr be the size of F, where p is prime, the size of the 2-Sylow subgroup of SL2(F) is 2n, where n = ord2(p2 − 1) + ord2(r).

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