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  1. #1
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    Is the symmetric group S[/SIZE]3 a direct product of nontrivial groups?


    could anyone help me ?? please?
    (what does it mean by 'direct product of nontrivial groups'?)
    Last edited by jin_nzzang; August 29th 2009 at 11:44 PM.
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  2. #2
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by jin_nzzang View Post
    [snip]
    (what does it mean by 'direct product of nontrivial groups'?)
    A "trivial group" is a group with one element.

    This page:
    Definition:Internal Direct Product - ProofWiki
    may help to remind you what a "direct product" is.
    Last edited by mr fantastic; August 30th 2009 at 08:25 PM. Reason: Added quote
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  3. #3
    ynj
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    Quote Originally Posted by jin_nzzang View Post
    Is the symmetric group S[/size]3 a direct product of nontrivial groups?
    [snip]
    No. Else it would only be the direct product of groups of order 2,3. but any groups with order 2 is isomorphic to Z_2 and any groups with order 3 is isomorphic to Z_3. That means S_3\cong Z_2\times Z_3\cong Z_6,which is cyclic and clearly impossible.
    Last edited by mr fantastic; August 30th 2009 at 08:26 PM. Reason: Added quote
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  4. #4
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    Or:

    If S_3 = H' \times K', then you would be able to find two subgroups H, K of S_3 such that:
    1) H is isomorphic to H', K is isomorphic to K'
    2) H is a normal subgroup of S_3, K is a normal subgroup of S_3
    3) HK = S_3
    4) H \cap K = 0.

    And this is clearly impossible since only nontrivial normal subgroup of S3 is the one generated by 3-cycle.
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