# Thread: Subgroup Proof

1. ## Subgroup Proof

Let G=GL(2,R) and let K be a subgroup of R*. Prove if H = {AeG | det(A)eK} show H is a subgroup of G. (Note: e means element of)

Here's my attempt. Let A,BeH then A,BeG so det(A),det(B)eK. Since K is a subgroup of R* then det(A)det(B)eK. Then since det(A)det(B)eK then ABeG and so ABeH.

Let BeH. Then BeG and det(B)eK. Since BeG then B^(-1)eG and so B^(-1)eH.

This doesn't seem right, so any help appreciated.

2. ## Re: Subgroup Proof

Originally Posted by JSB1917
Let G=GL(2,R) and let K be a subgroup of R*. Prove if H = {AeG | det(A)eK} show H is a subgroup of G. (Note: e means element of)

Here's my attempt. Let A,BeH then A,BeG so det(A),det(B)eK. Since K is a subgroup of R* then det(A)det(B)eK. Then since det(A)det(B)eK then ABeG and so ABeH.

Let BeH. Then BeG and det(B)eK. Since BeG then B^(-1)eG and so B^(-1)eH.

This doesn't seem right, so any help appreciated.
This isn't quite right. Try to prove the more general result that if $\displaystyle f:G\to H$ is a group homomorphsim and $\displaystyle K\leqslant H$ then $\displaystyle f^{-1}(K)\leqslant G$.