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Characteristic Subgroup is Normal

  1. Jun 15, 2008 #1
    The problem statement, all variables and given/known data
    Recall that a subgroup N of a group G is called characteristic if f(N) = N for all automorphisms f of G. If N is a characteristic subgroup of G, show that N is a normal subgroup of G.

    The attempt at a solution
    I must show that if g is in G, then gN = Ng. Let n be in N. Since N is characteristic, there are automorphisms f and f' of G and elements a and a' of G such that n = f(a) = f'(a'). There are also elements b and b' in G such that g = f(b) = f'(b') so that gn = f(b)f(a) = f(ba) = f'(b')f(a) = f(b)f'(a') = f'(b')f'(a') = f'(b'a'). This is all I can think of and I don't see how this allows me to prove that N is normal. Am I missing something?
     
  2. jcsd
  3. Jun 15, 2008 #2

    matt grime

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    A subgroup is normal if and only if it is fixed (set wise) under conjugation by any element. That's precisely what gN=Ng (i.e gNg^-1 = N) says
     
  4. Jun 15, 2008 #3

    quasar987

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    Wouldn't it be nice if f_g:G-->G: h-->ghg^-1 were an automorphism of G for each g in G?
     
  5. Jun 15, 2008 #4
    Thanks for the tip quasar987. I understand now.
     
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