Matrix and Quaternion Groups

The dihedral and symmetric groups come from geometry and combinatorics. A third source is linear algebra: the invertible matrices over a field form a group under multiplication, and their subgroups supply many of the standard examples of infinite and finite noncommutative groups. Alongside them sits a small exceptional group, the quaternion group $Q_8$, which has order $8$ like the dihedral group $D_8$ but is structurally different — a difference that a subgroup diagram makes visible.

Fields, in brief

Matrix entries are drawn from a field. A full treatment waits for field theory; here only the definition is needed.

A field is the smallest setting in which $+$, $-$, $\times$, and division by nonzero elements all make sense. The examples for now are $\mathbb{Q}$, $\mathbb{R}$, and ${\mathbb{F}}_p=\mathbb{Z}/p\mathbb{Z}$ for a prime $p$ — the last a finite field, whose additive and multiplicative structure was assembled in modular arithmetic. Write $F^{\times }=F\setminus \{0\}$ for the multiplicative group of a field.

The general linear group

Three facts from linear algebra, valid over any field with the same formulas as over $\mathbb{R}$, make this a group.¹

• Closure. The determinant is multiplicative, $\det (AB)=\det (A)\det (B)$, so if $\det A\ne 0$ and $\det B\ne 0$ then $\det (AB)\ne 0$.
• Associativity. Matrix multiplication is associative.
• Inverses. A square matrix is invertible if and only if its determinant is nonzero, and the inverse (by the adjugate formula) again has entries in $F$ and nonzero determinant, so $A^{-1}\in {\mathrm{GL}}_n(F)$.

The identity is the identity matrix $I$. The group is nonabelian for every $n\ge 2$, since matrix products generally depend on order.

Determinant one: the special linear group

The determinant is itself a multiplicative map ${\mathrm{GL}}_n(F)\to F^{\times }$, and the matrices it sends to $1$ form a distinguished subgroup.

The description matrices the determinant sends to $1$ says that ${\mathrm{SL}}_n(F)$ is the kernel of the determinant, viewed as a homomorphism$\det :{\mathrm{GL}}_n(F)\to F^{\times }$. Kernels of homomorphisms are subgroups automatically, which is a cleaner reason for the closure than checking products by hand; the general theory is developed in the isomorphism theorems.

Geometrically, over $\mathbb{R}$ the group ${\mathrm{SL}}_n(\mathbb{R})$ consists of the linear maps that preserve signed volume: ${\mathrm{SL}}_2(\mathbb{R})$ preserves the area of every parallelogram, and ${\mathrm{SL}}_3(\mathbb{R})$ the volume of every parallelepiped.² A general element of ${\mathrm{GL}}_n(\mathbb{R})$ distorts the grid and rescales area by the factor $\det A$; the special linear subgroup consists of the shears and rotations that leave the scale alone.

Orthogonal groups

A second subgroup of ${\mathrm{GL}}_n(\mathbb{R})$ collects the length-preserving maps.

From $A^{\mathsf{T}}A=I$ and multiplicativity, $\det (A{)}^2=1$, so every orthogonal matrix has determinant $\pm 1$.² The determinant-one elements $SO(n)=O(n)\cap {\mathrm{SL}}_n(\mathbb{R})$ are the rotations; in the plane, $SO(2)$ is the circle of rotation matrices, and the full $O(2)$ adds the reflections. The dihedral group appears here concretely: $D_{2n}$ is the finite subgroup of $O(2)$ generated by rotation through $2\pi /n$ and one reflection, matching the geometric definition from the dihedral group.

Order over a finite field

When $F$ is finite the general linear group is finite, and its order counts the ordered bases of $F^n$.

Idea. An invertible matrix is a choice of $n$ linearly independent columns. The first column is any nonzero vector: $q^n-1$ choices. The second is any vector outside the line spanned by the first: $q^n-q$ choices. The $k$-th is any vector outside the $(k-1)$-dimensional span already chosen: $q^n-q^{k-1}$ choices. Multiplying gives the formula.¹ Two small cases:

• $q=2,n=2$. $|{\mathrm{GL}}_2({\mathbb{F}}_2)|=(2^2-1)(2^2-2)=3\cdot 2=6$, and indeed ${\mathrm{GL}}_2({\mathbb{F}}_2)\cong S_3$, the smallest nonabelian group — the six invertible $2\times 2$ matrices over ${\mathbb{F}}_2$ permute the three nonzero vectors of ${\mathbb{F}}_2^2$.
• $q=3,n=2$. $|{\mathrm{GL}}_2({\mathbb{F}}_3)|=(9-1)(9-3)=8\cdot 6=48$, of which the determinant-one matrices number $|{\mathrm{SL}}_2({\mathbb{F}}_3)|=48/|{\mathbb{F}}_3^{\times }|=48/2=24$, since $\det$ maps onto ${\mathbb{F}}_3^{\times }=\{1,2\}$ with each fiber the same size.

The quaternion group

The second small nonabelian group is not a matrix group by definition, though it embeds in one. It is given by an explicit table.

The three imaginary units cycle $i\to j\to k\to i$ under multiplication when taken in order, and reverse to a sign change when taken backward: $ji=-k$, $kj=-i$, $ik=-j$. The element $-1$ commutes with everything and is the unique element of order $2$; the six elements $\pm i,\pm j,\pm k$ all have order $4$.

Like the dihedral group, $Q_8$ is generated by two elements with a short list of relations. Taking $i$ and $j$ as generators,

with $k=ij$ and $-1=i^2$ recovered from the generators.¹ The relation $ji{j}^{-1}=i^{-1}$ mirrors the dihedral $sr{s}^{-1}=r^{-1}$, so the two groups of order $8$ are built from similar-looking presentations; the difference is the single extra relation $i^2=j^2$, which forces $j^2\ne 1$ and thereby removes all the order-$2$ reflections that $D_8$ has.

Cyclic subgroups and orders

Each imaginary unit generates a copy of the cyclic group of order $4$. Following the powers of $i$, for instance, gives a four-step loop that visits $1$, $i$, the central element $i^2$, and $i^3$, returning to $1$.

Both $Q_8$ and $D_8$ have order $8$ and are nonabelian, yet they are not isomorphic. The cleanest distinction is in their elements of order $2$: the dihedral group $D_8$ has five (the central rotation by a half-turn and four reflections), while $Q_8$ has exactly one (the central element $-1$). Every subgroup of $Q_8$ except the trivial one therefore contains $-1$, so the subgroup diagram of $Q_8$ funnels down to a single point above the bottom, whereas $D_8$ fans out into several minimal subgroups.

Both groups embed in matrix groups, which is where the matrix and quaternion threads rejoin. The dihedral group $D_{2n}$ sits inside ${\mathrm{GL}}_2(\mathbb{R})$ as the rotation and reflection matrices of the $n$-gon, and $Q_8$ sits inside ${\mathrm{GL}}_2(\mathbb{C})$ under the map

Every group can be realized this way — as symmetries of some structure or as matrices acting on a space — through homomorphisms and group actions and, in the extreme, Cayley's theorem.

The two tables of order counts differ, so no relabeling can turn one group into the other. Counting elements of each order is the first and cheapest test for whether two finite groups are the same.