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Math Help - Normal Subgroup

  1. #1
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    Normal Subgroup

    1. Find all the normal subgroups of <a> of S3 and <b> of D4, where a is in S3 and b is in D4. I know A4 is a normal subgroup of S3. Are there any others?
    2. Let H be a subgroup of an arbitrary group G. Prove H is normal iff it has the following property: For all a,b, in G, ab is in H iff ba is in H.
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  2. #2
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    We can think of S_3 as \{e,a,a^2.b,ab,a^2b\} with the relation ba=a^2b.
    (Here e=(1),a=(1,2,3),b=(1,2)).
    And here is the subgroup diagram.
    Attached Thumbnails Attached Thumbnails Normal Subgroup-picture.gif  
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  3. #3
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    We can think of D_4 as \{e,a,a^2,a^3,b,ab,a^2b,a^3b\} with the relation ba=a^3b. (Here e=(1),a=(1,2,3,4),b=(2,3).
    And here is the subgroup (much nicer looking).
    Attached Thumbnails Attached Thumbnails Normal Subgroup-picture.gif  
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