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Math Help - Subgroups

  1. #1
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    Subgroups

    I am confused about the definition of subgroups.

    I am looking at matrices of the form [a 0 ; 0 0] with a not equal to 0. This is a subset of GLn2

    I thought this was a subgroup but my book says it's not. I figured that any matrix of that form multiplied by [1 0; 0 0] gives back the matrix, so [1 0; 0 0] is the identity and is in the group. Also, any matrix of that form multiplied [1/a 0; 0 0] gives the previously found identity.

    Does the identity of the subgroup need to be the identity of the larger group (in this case [1 0; 0 1]) or the identity that works for the subgroup?
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  2. #2
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    Re: Subgroups

    The identity must be the same, so those matrices have no inverses.

    You may prove it like this: If H is a subgroup of G, and x i in H, we have xx^{-1}=1_G and also xx^{-1}=1_H which implies that 1_G=1_H
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  3. #3
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    Re: Subgroups

    I see. Thanks for the answer.
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  4. #4
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    Re: Subgroups

    I can see how this would be confusing.

    The set you gave is in fact a group in its own right (which is isomorphic to the field from which these matrices take their entries). The problem is that it doesn't get its information from GL_2(F); in fact, none of the elements in this set are actually in this group.
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