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Math Help - Subgroups

  1. #1
    Newbie
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    Jan 2009
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    Subgroups

    Hi,

    Using the subgroup test, or otherwise determine which of the following subsets H are subgroups of the stated group G, giving ur reasons:

    a) H= {e, (12), (23), (13)} G= S3 (S sub 3)

    b) H= {1, x, x^2, ...x^7} G= {1, x, x^2, ..., x^8}, the cyclic group of order 9

    c) H= {A | det(A) is in Z} G=GL3(R) (L3= L sub 3, Z=integers)

    d) H= {[k] | k even} G= Z/13Z, with group operation being addition.

    Could u pls get me started.
    Thanks!
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  2. #2
    Junior Member
    Joined
    Dec 2009
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    H is not subgroup

    a) H is not subgroup of S(3)

    Sinse (1 2)(2 3)=(1 2 3) which does not belong to H, so the closure low is not satisfied
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    Champaign, Illinois
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    Quote Originally Posted by choo View Post
    Hi,

    Using the subgroup test, or otherwise determine which of the following subsets H are subgroups of the stated group G, giving ur reasons:

    a) H= {e, (12), (23), (13)} G= S3 (S sub 3)

    b) H= {1, x, x^2, ...x^7} G= {1, x, x^2, ..., x^8}, the cyclic group of order 9

    c) H= {A | det(A) is in Z} G=GL3(R) (L3= L sub 3, Z=integers)

    d) H= {[k] | k even} G= Z/13Z, with group operation being addition.

    Could u pls get me started.
    Thanks!
    b.)  x,x^7\in H \implies x^8 \in H ...

    d.) If  H was a group, then every element has an additive inverse. But  2 for example has no additive inverse since  11 \not\in H .
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