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Abstract Algebra Questions .

  1. Nov 1, 2011 #1
    Abstract Algebra Questions.....

    I have two problems that I'm a little puzzled by, hopefully someone can shed some light.

    1) Show that if H and K are subgroups of the group G, then H U K is closed under inverses.

    2) Let G be a group, and let g ε G. Define the centralizer, Z(g) of g in G to be the subset
    Z(g) = {x ε G | xg = gx}.
    Prove that Z(g) is a subgroup of G.
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    For problem 2 this is what I have but I am not sure if it is correct.

    Since eg = ge for g in G, we know Z(g) is not the empty set.

    -Take a in Z(g) and b in Z(g), and take any g in G, then we have...
    (ab)g = a(bg) = a(gb) = (ag)b = (ga)b = g(ab). Thus ab is in Z(g).

    - Take a in Z(g) and g in G. Then we know....
    ag = ga
    (a^-1* a )g = (a^-1 * g) a (multiplying both sides by a inverse)
    e * g = a^-1 * g*a
    g * a^-1 = a^-1 * g * (a * a^-1) ( multiplying again by a invese)
    g * a^-1 = a^-1 * g

    Thus a^-1 is in Z(g), so Z(g) is a subgroup of G.
     
  2. jcsd
  3. Nov 2, 2011 #2

    Deveno

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    Science Advisor

    Re: Abstract Algebra Questions.....

    what you did on 2 is fine. you could have saved a little time by showing b-1 is in Z(g) whenever b is, and then showing ab-1 is in Z(g) when a and b are, but not much.

    for 1) x in HUK means:

    x is in H...or
    x is in K..or both.

    so start by assuming x is in H, what can you say about x-1?

    next, if x is not in H, it must be in K, and use a similar agument.
     
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