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Math Help - Subgroup Proof

  1. #1
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    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.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Subgroup Proof

    Quote Originally Posted by JSB1917 View Post
    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 f:G\to H is a group homomorphsim and K\leqslant H then f^{-1}(K)\leqslant G.
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