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Math and science::Algebra::Aluffi

Subgroups of the general linear group

Below are three subgroups of GLn(R), the group of invertible n×n matrices with real entries.

SLn(R)
The subset of GLn(R) consisting of matrices with determinant 1. "S" is short for "special".
On(R)
The subset of GLn(R) consisting of matrices M such that MTM=MMT=In. "O" here is short for "orthogonal".
SOn(R)
The subset of On(R) consisting of matrices with determinant 1.

Can you proof that these 3 subsets form subgroups?


To prove that a subset of a group is a subgroup, we need to show that:

  1. It contains the identity element.
  2. It is closed under the group operation.
  3. It is closed under inverses.

Orthogonal group vs. special orthogonal group

The orthogonal group has the requirement that MTM=MMT=In. This means that the columns of M are orthogonal to each other and that they all have length 1. This is quite a strong requirement and you may wonder what is left to distinguish this group from the special orthogonal group. The answer is that the special orthogonal group has the additional requirement that the determinant of M is positive 1. What is not in the special orthogonal group are all the matrices with determinant -1.

SLn(R)

SLn(R) is a subgroup of GLn(R).

Proof. The identity matrix for GLn(R) has determinant 1, so it is in SLn(R). For any matrices A and B in Gn(R, det(AB)=det(A)det(B). An so if det(A)=det(B)=1, then det(AB)=1. So AB is in SLn(R). As AA1=In, if A has determinant 1, then so too must A1. So we have closure under matrix multiplication and inverses, and the identity is present.

On(R)

SOn(R)

SOn(R) is a subgroup of On(R).

Proof idea. An orthogonal matrix is one whose columns are orthogonal to each other. The main task is to show that the matrix multiplication of two orthogonal matrices is also orthogonal. Let A and B be orthogonal matrices and consider AB=C. Let's be concrete and fix the matrix dimensions to be 3×3. Each column of C is a linear combination of the columns of A with coefficients given by the columns of B. Consider testing the dot product of two columns c1 and c2 of C. c1 will be B11a1+B12a2+B13a3 where ax is the xth column of A. Similarly, c2 will be B21a1+B22a2+B23a3. When we take the dot product of c1 and c2, out of the 9 sub-products that make up the dot product, the only terms that will survive are those like B11a1B21a1, as the other terms will be orthogonal. But B11a1B21a1 is just B11B21a1, and so the dot product of c1 and c2 is just [B11B21a1B12B22a2B13B23a3].

Now remember that AAT=I3 which is another way of saying that the dot product of any column with itself is 1, and the dot product of any two different columns is 0. So a1=a2=a3=1 and so the dot product of c1 and c2 is [B11B21B12B22B13B23].

But this is just the dot product of the first column of B with the second column of B, which is 0. For the same two columns of C, the dot product will be 1. This is what we wanted in order for C to be orthogonal.

Proof.

If G1 and G2 are subgroups of a group G, then their intersection G1G2 is also a subgroup of G. Applying this rule to SLn(R) and On(R) as subgroups of GLn(R), we see that SOn(R) is a subgroup of On(R) also.