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Problem about normal subgroups

  1. Jun 6, 2012 #1
    1. The problem statement, all variables and given/known data

    If A,B and C are normal subgroups of G where B[itex]\subseteq[/itex]A show that
    A[itex]\bigcap[/itex]BC=B(A[itex]\bigcap[/itex]C)

    2. Relevant equations

    3. The attempt at a solution
    Let x[itex]\in[/itex]A[itex]\bigcap[/itex]BC.then x[itex]\in[/itex]A and x[itex]\in[/itex]BC
    Now as B[itex]\subseteq[/itex]A thus BA=A.thus left side is BA[itex]\bigcap[/itex]BC

    Dont know how to proceed.
     
  2. jcsd
  3. Jun 6, 2012 #2

    micromass

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    So you know that [itex]x\in A[/itex] and [itex]x\in BC[/itex]. The last means that there are [itex]b\in B[/itex] and [itex]c\in C[/itex] such that [itex]x=bc[/itex].

    You must prove that [itex]x\in B(A\cap C)[/itex]. So you must write x as a product of something in B and something in [itex]A\cap C[/itex].
     
  4. Jun 6, 2012 #3
    x=bc [itex]\Rightarrow[/itex] a-1bac[itex]\in[/itex]BC
    Also a-1bca[itex]\in[/itex]BC Now a-1b=a1 for some a1[itex]\in[/itex]A[itex]\Rightarrow[/itex]a1ca[itex]\in[/itex]BC
    a1(aa-1)ca[itex]\in[/itex]BC
    (a1a)c1[itex]\in[/itex]BC for some c1[itex]\in[/itex]C
    a2c1[itex]\in[/itex]BC
    Cant still figure out
     
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