1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    User Avatar
    Science Advisor
    Homework Helper

    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook