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Abstract Algebra homework Direct products

  1. Nov 15, 2012 #1
    1. The problem statement, all variables and given/known data

    We've shown if G[itex]_{1}[/itex],G[itex]_{2}[/itex],...,G[itex]_{n}[/itex] are subgroups of G s.t.

    1)G[itex]_{1}[/itex],G[itex]_{2}[/itex],...,G[itex]_{n}[/itex] are all normal
    2)Every element of G can be written as g[itex]_{1}[/itex]g[itex]_{2}[/itex]...g[itex]_{n}[/itex] with g[itex]_{i}[/itex][itex]\in[/itex]G
    3)For 1[itex]\leq[/itex]i[itex]\leq[/itex]n, G[itex]_{i}[/itex][itex]\cap[/itex]G[itex]_{1}[/itex],G[itex]_{2}[/itex],...,G[itex]_{i-1}[/itex]=e

    then G[itex]\cong[/itex]G[itex]_{1}[/itex]xG[itex]_{2}[/itex]x...xG[itex]_{n}[/itex]

    Show, by example, that if we replace 3) with the condition G[itex]_{i}[/itex][itex]\cap[/itex]G[itex]_{k}[/itex]=e for all i[itex]\neq[/itex]k then G does not need to be isomorphic to G[itex]_{1}[/itex]xG[itex]_{2}[/itex]x...xG[itex]_{n}[/itex]



    2. Relevant equations



    3. The attempt at a solution

    I tried to find an example with abelian groups like[itex]Z[/itex][itex]_{60}[/itex], but nothing seemed to work. Now I'm trying groups that are themselves direct products, but I seem to be missing the big picture.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 16, 2012 #2

    micromass

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    You need to find a good homomorphism

    [tex]\varphi:G_1\times...\times G_n\rightarrow G[/tex]

    and show that that is an isomorphism. What do you think you can choose as [itex]\varphi[/itex]??
     
  4. Nov 16, 2012 #3

    micromass

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    You also might want to think about an induction on n. Can you show it for n=2??
     
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