Special linear group

In terms of natural numbers
Let $$n$$ be a natural number and $$k$$ a field. The special linear group of degree $$n$$ over $$k$$ is defined in the following equivalent ways:


 * 1) It is the group of all invertible matrices of order $$n$$, having determinant 1.
 * 2) It is the kernel of the determinant homomorphism from the defining ingredient::general linear group of degree $$n$$ over $$k$$ to the multiplicative group of $$k$$.

As a functor from fields to groups
If we fix $$n$$, we can think of $$SL_n$$ as a functor from the category of fields to the category of groups.

As an IAPS
For a fixed field $$k$$, the general linear groups $$SL(n,k)$$ form an IAPS of groups parametrized by $$n$$. In other words, we naturally have concatenation maps:

$$\Phi_{m,n}: SL(m,k) \times SL(n,k) \to SL(m+n,k)$$

This map takes a matrix $$A$$ of order $$m$$ and a matrix $$B$$ of order $$n$$ and putputs the block diagonal matrix with blocks $$A$$ and $$B$$.

As a functor from fields to IAPSes
If we fix neither $$n$$ nor $$k$$, we can view $$SL$$ as a functor from fields to the category of IAPSes of groups.

Subgroups

 * Subgroup::Special orthogonal group
 * Subgroup::Symplectic group
 * Subgroup::Special orthogonal similitude group
 * Subgroup::Unipotent upper-triangular matrix group

Supergroups

 * Supergroup::General linear group
 * Supergroup::Special affine group (also called the affine special linear group)
 * Supergroup::Outer special linear group
 * Supergroup::Special semilinear group

Group and subgroup operations

 * Intersection with the orthogonal group gives the special orthogonal group.
 * Intersection with the orthogonal similitude group gives the special orthogonal similitude group.
 * Normalizer in the general linear group is the whole general linear group.

Generating sets for the special linear group

 * Elementary matrices generate the special linear group: This is true for all special linear groups over all fields.
 * Dickson's theorem: This shows that $$SL(2,q)$$ is practically always generated by any upper-triangular unipotent mtarix and any lower-triangular unipotent matrix.

Presentations for the special linear group

 * Steinberg presentation for the special linear group: This is a presentation that works for special linear groups over fields. Over rings that are not fields, the presentation gives a group that has a homomorphism to the special linear group that need not in general be either injective or surjective.

Finite fields
Here are some general facts:


 * Special linear group is quasisimple for $$n \ge 2$$, except the case that $$n = 2$$ and $$k$$ has two or three elements. Thus, all the corresponding general linear groups have a simple non-abelian subquotient. In the case that the field has characteristic two, the general linear group coincides with the special linear group, and it is centerless, so it turns out to be a simple non-abelian group itself.

GAP implementation
The GAP command for constructing the special linear group is GAP:SpecialLinearGroup. It can be invoked using either SpecialLinearGroup or SL. It takes two arguments, the first of which is the degree (i.e., the order of matrices) and the second is either a ring or a prime power (for which the corresponding field is considered the ring). For instance, to define $$SL(2,3)$$, we write:

G := SL(2,3);