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Math Help - Klein 4-group

  1. #1
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    Klein 4-group

    Let H be the Klein 4-group. To which familiar group is the quotient group
    S4/H isomorphic? Justify your answer.

    Any suggestions or hints would be fabulous!
    Last edited by mathgirl1188; October 24th 2010 at 11:10 AM.
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  2. #2
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    My only suggestion would be to sit down and do it! S_4, the group of permutations on 4 "symbols", contains 4!= 24 elements. It is tedious but not impossible to write out its operation table. It has a subgroup isomorphic to the Klein group (if it didn't this problem would make no sense). Can you identify that subgroup? S_4/H has 8 members. Can you work out the operation table for it from the operation table for S_4?
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    My only suggestion would be to sit down and do it! S_4, the group of permutations on 4 "symbols", contains 4!= 24 elements. It is tedious but not impossible to write out its operation table. It has a subgroup isomorphic to the Klein group (if it didn't this problem would make no sense). Can you identify that subgroup? S_4/H has 8 members. Can you work out the operation table for it from the operation table for S_4?

    A little typo slip there: S_4/H has 6 elements, of course, and since there are only two different groups,

    up to isomorphism, of order 6, there are no many options from where to choose...

    Now, remember that a quotient group is abelian iff the normal subgroup contains the derived, or commutator, subgroup

    of the whole group, which is here A_4 , so...

    Tonio
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  4. #4
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    Oops, yes, 24/4= 6, not 8! I really need to work on my arithmetic!
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