Answer to Question #210406 in Abstract Algebra for Maheen Fatima

Question

Prove that SL(n,F) is a normal subgroup of GL(n,F)?

Answer 1

Recall:

GL(n,R) is the collection of the general

n×n invertible matrices.

SL(n,R)={X∈GL(n,R)∣det(X)=1}.

To prove: SL(n,R) is a normal subgroup of G

Let X∈SL(n,R)

and let P∈G

Then we have

det(PXP-¹)=det(P)det(X)det(P)^-1=det(X)=1

and hence the conjugate PXP^-1

PXP^-1 is in SL(n,R)

Therefore, SL(n,R)

SL(n,R) is a normal subgroup of G