Special linear group: Difference between revisions

Revision as of 16:41, 16 March 2009

This article defines a natural number-parametrized system of algebraic matrix groups. In other words, for every field and every natural number, we get a matrix group defined by a system of algebraic equations. The definition may also generalize to arbitrary commutative unital rings, though the default usage of the term is over fields.
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Definition

In terms of natural numbers

Let $n$ be a natural number and $k$ a field. The special linear group of order $n$ over $k$ is defined as the group of all invertible matrices of order $n$ , having determinant 1.

As a map

As a functor from fields to groups

If we fix $n$ , we can think of $S L_{n}$ as a functor from the category of fields to the category of groups.

As an IAPS

Further information: SL IAPS

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

$Φ_{m, n} : S L (m, k) \times S L (n, k) \to S L (m + n, k)$

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 $S L$ as a functor from fields to the category of IAPSes of groups.

Relation with other linear algebraic groups

Subgroups

Supergroups

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.

Particular cases

Finite fields

Here are some general facts:

Special linear group is quasisimple for $n \geq 2$ , 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.

Size of field	Order of matrices	Common name for the special linear group	Order of group	Comment
q	1	Trivial group	$1$	Trivial
2	2	Symmetric group:S3	$6 = 2 \cdot 3$	supersolvable but not nilpotent
3	2	Special linear group:GL(2,3)	$24 = 2^{3} \cdot 3$	solvable but not supersolvable
4	2	Alternating group:A5	$60 = 2^{2} \cdot 3 \cdot 5$	simple non-abelian group
5	2	Special linear group:SL(2,5)	$240 = 2^{4} \cdot 3 \cdot 5$	quasisimple, with inner automorphism group isomorphic to alternating group:A5.
2	3	General linear group:GL(3,2)	$168 = 2^{3} \cdot 3 \cdot 7$	simple non-abelian group

@@ Line 6: / Line 6: @@
 Let <math>n</math> be a [[natural number]] and <math>k</math> a [[field]]. The '''special linear group''' of order <math>n</math> over <math>k</math> is defined as the group of all invertible matrices of order <math>n</math>, having determinant 1.
+==As a map==
+===As a functor from fields to groups===
+If we fix <math>n</math>, we can think of <math>SL_n</math> as a functor from the category of fields to the [[category of groups]].
+===As an IAPS===
+{{further|[[SL IAPS]]}}
+For a fixed field <math>k</math>, the general linear groups <math>SL(n,k)</math> form an [[IAPS of groups]] parametrized by <math>n</math>. In other words, we naturally have concatenation maps:
+<math>\Phi_{m,n}: SL(m,k) \times SL(n,k) \to SL(m+n,k)</math>
+This map takes a matrix <math>A</math> of order <math>m</math> and a matrix <math>B</math> of order <math>n</math> and putputs the block diagonal matrix with blocks <math>A</math> and <math>B</math>.
+===As a functor from fields to IAPSes===
+If we fix neither <math>n</math> nor <math>k</math>, we can view <math>SL</math> as a functor from fields to the category of [[IAPS of groups|IAPSes of groups]].
+==Relation with other linear algebraic 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]]
+===Group and subgroup operations===
+* [[Intersection of subgroups|Intersection]] with the [[orthogonal group]] gives the [[special orthogonal group]].
+* [[Intersection of subgroups|Intersection]] with the [[orthogonal similitude group]] gives the [[special orthogonal similitude group]].
+* [[Normalizer]] in the general linear group is the whole general linear group.
+==Particular cases==
+===Finite fields===
+Here are some general facts:
+* [[Special linear group is quasisimple]] for <math>n \ge 2</math>, except the case that <math>n = 2</math> and <math>k</math> 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.
+{| class="wikitable" border="1"
+!Size of field !! Order of matrices !! Common name for the special linear group !! Order of group !! Comment
+|-
+| q || 1 || [[Trivial group]] || <math>1</math> || Trivial
+|-
+| 2 || 2 || [[Symmetric group:S3]] || <math>6 = 2 \cdot 3</math> || [[Supersolvable group|supersolvable]] but not [[nilpotent group|nilpotent]]
+|-
+| 3 || 2 || [[Special linear group:GL(2,3)]] || <math>24 = 2^3 \cdot 3</math> || [[solvable group|solvable]] but not supersolvable
+|-
+| 4 || 2 || [[Alternating group:A5]] || <math>60 = 2^2 \cdot 3 \cdot 5</math> || [[simple non-abelian group]]
+|-
+| 5 || 2 || [[Special linear group:SL(2,5)]] || <math>240 = 2^4 \cdot 3 \cdot 5</math> || [[quasisimple group|quasisimple]], with inner automorphism group isomorphic to [[alternating group:A5]].
+|-
+| 2 || 3 || [[General linear group:GL(3,2)]] || <math>168 = 2^3 \cdot 3 \cdot 7</math> || [[simple non-abelian group]]
+|-
+|}