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Math Help - Simple Abelian Group

  1. #1
    MHF Contributor chiph588@'s Avatar
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    Simple Abelian Group

    Could someone explain why  G being simple and abelian implies it's cyclic?

    Thanks!
    -Chip
    Last edited by chiph588@; April 11th 2010 at 05:05 PM.
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  2. #2
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    Quote Originally Posted by chiph588@ View Post
    Could someone explain why  G simple and abelian implies it is cyclic?

    Thanks!
    -Chip
    choose 1 \neq x \in G. then H=\langle x \rangle is a non-trivial normal subgroup of G because G is abelian. so H=G because G is simple.
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by NonCommAlg View Post
    choose 1 \neq x \in G. then H=\langle x \rangle is a non-trivial normal subgroup of G because G is abelian. so H=G because G is simple.
    Ah yes this proof was rather... *takes glasses off* simple.
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  4. #4
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    Quote Originally Posted by chiph588@ View Post
    Ah yes this proof was rather... *takes glasses off* simple.
    G should be finite though.
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  5. #5
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    Quote Originally Posted by aliceinwonderland View Post
    G should be finite though.
    of coures, infinite abelian group are never simple. also, obviously, the finite ones are either of order 1 or a prime number.
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