The Special and General Linear Groups: SLn(F) & GLn(F)

Joseph Heavner

January 12, 2016

Definition: Let F be a field, typically R or C under their usual operations, then theGeneral Linear Group, GLn(F), is the group, under matrix multiplication, of non-singular n n matrices with entries in the field F.

Let us prove that this is indeed a group. To do this, we must simply go back tothe group axioms, i.e. the definition of a group. Recall the four group axioms:

Closure: If a, b G, then ab G) Associativity: For any a, b, c G (ab)c = a(bc) Identity: There exists an element e G such that for any a G ae = ea = a.

(This is easily shown to be unique.)

Inverse: For all a G there exists an inverse, a1 such that aa1 = a1a = e.(This is similar to the identity in that it is easily proven to be unique for anygiven element.)

One should make special note that there is no commutative property. Commutativegroups are a special class of groups commonly known as Abelian groups.

So, to prove that GLn(F) is indeed a group is quite easy.

Proof: Because we are working over a field, we have guaranteed closure of matrixmultiplication. Matrix multiplication is associative. The identity element is simplythe matrix In. And, because all our matrices are invertible, for any member A wehave another member A1 such that AA1 = A1A = In.

One could be more formal about this, but the exercise is rather simple.

A particularly interesting problem with GLn(F) is to determine the size of thegroup. If your field is infinite, indeed so is the size of your group, but for finite fields,one can prove (and we will not do so here) that the number of elements in GLn(Fq) is

n1k=1

(qn qk)

Interestingly, both GLn(F) and SLn(F) are Lie groups, making them of particularinterest, as a Lie group is on the intersection of algebra and geometry (or topology).

Definition: A Lie Group is a group that also forms a differentiable manifold (a man-ifold whose transition maps are differentiable), the same kind that one does calculuson.

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Definition: Let F be a field, again usually over the reals or complex numbers, thenthe Special Linear Group, SLn(F), is the group, under matrix multiplication, of alln n matrices with determinant 1 whose entries are in the field F.

Again, let us prove that this is indeed a group. This time, we will not resort togoing axiom by axiom.

Proof: The special linear group is actually a normal subgroup of the general lineargroup. Because we only care that SLn(F) is a group at all, we shall prove that it is asubgroup of the general linear group.

Because the determinant of the matrices in the group are non-zero, they are invertible.Thus, the set associated with the special linear group is a subset of that associatedwith the general linear group. Now, suppose a, b SLn(F), then, because A has aninverse, we have that A1 GLn(F). The determinant of the inverse is the inverse ofthe determinant, i.e.

det(A1) =1

det(A)

Hence, by definition of the groups elements having determinant 1, we have thatdet(A1) = 1. So, A1 SLn(F). Because the special linear group is non-empty,closed under matrix multiplication, and closed under inverses, and we have that theassociated set is a subset of the general linear groups associated set, we have suffi-ciently tested our hypothesis.

The test used above is sometimes given a special name like the two part sub-group test. It is not very exciting, but it recognizes that we need not check associa-tivity due to the property being inherited in the subgroup, nor do we need to worryabout the identity, as that is handled by the conditions regarding closure and inverses.

Example: A particularly interesting example of the special linear group is SL2(R).This group is explicitly defined by (with representing matrix multiplication):

SL2(R) =({(a bc d

): a, b, c, d R , ad bc = 1},

)This is arguably most important in physics, though its importance in algebra

(particularly representation theory) and geometry (particularly topology) cannot beignored. In physics, this group comes up fairly regularly, for instance the three dimen-sional Lorentz group with metric signature (,+,+) is SL2(R). The Lorentz Groupis the group of all Lorentz transformations (transformations that account for the con-stancy of the speed of light, ultimately leading to time dilation, length contraction,and the correspondence principle, to name a few things) in Minkoswki space (a fourdimensional manifold used in Einsteins theory of relativity to represent space)

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