Answer to Question #210412 in Abstract Algebra for Maheen Fatima

Question

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

Answer 1

Let us prove that $SL(n,F)=\{A\in GL(n,F):\det A =1\}$ is a subgroup of $GL(n,F).$ If $A,B\in SL(n,F),$ then $\det A=\det B =1.$ It follows that $\det(AB)=\det A\cdot \det B=1\cdot 1=1,$ and hence $AB\in SL(n,F).$ It follows also that $\det (A^{-1})=\frac{1}{\det A}=\frac{1}{1}=1,$ and hence $A^{-1}\in SL(n, F).$ We conclude that $SL(n,F)$

is a subgroup of $GL(n,F).$