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Math Help - Normal Subgroups and Factor Groups

  1. #1
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    Normal Subgroups and Factor Groups

    Let G = GL(2,R) and let K be a subgroup of R*. Prove that H = {A ∈ G| det A ∈ K} is a normal subgroup of G.
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  2. #2
    Super Member Gamma's Avatar
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    Normal

    A subgroup H\leq G is said to be normal iff \forall g \in G we have gHg^{-1} \subset H

    So take A\in G=GL(2, \mathbb{R}) it is a group and therefore invertible so A^{-1} \in G but consider for any X\in H we would have:

    det(AXA^{-1})=det(A)det(X)det(A^{-1})=det(A)det(X)\frac{1}{det(A)}=det(X) so we see that for any X\in H and for all A\in G we get AXA^{-1}\in H.

    This shows AHA^{-1} \subset H thus it is normal.
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