Let G= GLn(R) in each of the cases when H is a normal subgroup identify the quotient group G/H. A) H=sLn(R) B)H=set of all upper triangular matrix C)H= set of all diagonal matrix?

Question

Answer

G/H in the case when H is a normal subgroup of G= GLn(R)

A) H = sLn(R)

In this case, we have H as the set of all scalar matrices in GLn(R). A scalar matrix is a diagonal matrix where all the diagonal entries are equal.

To find the quotient group G/H, we need to consider the left cosets of H in G. The left coset of H in G is defined as gH = {gh : h ∈ H}, where g is an element of G.

Claim: Every left coset of H in G contains exactly one scalar matrix.

Proof: Let g be an element of G. We need to show that the left coset gH contains exactly one scalar matrix.

Consider an arbitrary element gh in the left coset gH. Since H consists of scalar matrices, we can write h = kI, where k is a scalar and I is the identity matrix.

Now, if we multiply gh by any element of H, say h', we get (gh)h' = g(hh') = g(kIh') = (gkh')I = (kg')I, which is still a scalar matrix.

Therefore, every element in the left coset gH is a scalar matrix.

Now, let's consider two different elements g1 and g2 in G. We need to show that their left cosets g1H and g2H are either equal or disjoint.

Suppose there exists an element gh in both g1H and g2H. This means gh = g1h1 and gh = g2h2 for some h1 and h2 in H.

Then, we have g1h1 = g2h2, which implies g1 = g2(h2h1^(-1)). Since h2h1^(-1) is a scalar matrix, g1 and g2 are equal up to a scalar multiple.

Therefore, the left cosets g1H and g2H are either equal or disjoint.

Conclusion: The quotient group G/H consists of the left cosets of H in G, where each left coset corresponds to a unique scalar matrix. Therefore, the quotient group G/H is isomorphic to the set of all scalar matrices.

B) H = set of all upper triangular matrices

In this case, we have H as the set of all upper triangular matrices in GLn(R).

To find the quotient group G/H, we need to consider the left cosets of H in G.

Claim: Every left coset of H in G contains exactly one matrix in reduced row echelon form.

Proof: Let g be an element of G. We need to show that the left coset gH contains exactly one matrix in reduced row echelon form.

Consider an arbitrary element gh in the left coset gH. Since H consists of upper triangular matrices, we can write h = U, where U is an upper triangular matrix.

Now, if we multiply gh by any element of H, say h', we get (gh)h' = g(hh') = g(UU') = (gU)U', which is still an upper triangular matrix.

Therefore, every element

Answer

Quotient Group for Normal Subgroups in GLn(R)

A) H = sLn(R)
To find the quotient group G/H, we need to understand the structure of the normal subgroup H. In this case, H is the subgroup of GLn(R) consisting of all scalar matrices.

Quotient Group G/H:
The quotient group G/H is the set of all cosets of H in G, where each coset is represented by an element of G. Since H is a normal subgroup, the left cosets and right cosets of H in G coincide.

Structure of G:
The group G is the general linear group of n x n invertible matrices over the real numbers. It consists of all matrices A with non-zero determinant.

Cosets of H in G:
The cosets of H in G can be represented as:
G/H = {gH | g ∈ G}

Cosets:
Each coset gH is a set of matrices that are obtained by multiplying each matrix in H by the matrix g from G on the left.

Properties of Cosets:
1. The cosets of H form a partition of G, i.e., every element of G belongs to exactly one coset.
2. Two cosets g1H and g2H are either equal or disjoint.
3. The product of two cosets (g1H)(g2H) is equal to the coset (g1g2)H.

Example:
Suppose G is GL2(R), the general linear group of 2 x 2 invertible matrices over the real numbers. Let H be the subgroup consisting of all scalar matrices.

Structure of H:
H = {aI | a ∈ R, a ≠ 0}
where I is the identity matrix.

Cosets of H in G:
G/H = {gH | g ∈ G} = {g{aI} | g ∈ G, a ≠ 0}
= {gaI | g ∈ G, a ≠ 0}

Conclusion:
The quotient group G/H consists of all cosets of H in G, where each coset is represented by an element of G. The structure of H determines the structure of the cosets and their multiplication in the quotient group G/H.

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