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Homework Help: I can't tell the difference between these groups :/

  1. Feb 7, 2010 #1
    Z(G) = { x in G : xg=gx for all g in G } (center of a group G)

    C(g) = { x in G : xg=gx } (centralizer of g in G)



    I have to show both are subgroups, but what's the difference in the methods?

    To me the first set is saying all the elements x1, x2,... in G when composed with every element in g, commute.

    The second set tells me what x1, x2,... in G when composed with a single chosen element from g in G, commute.

    Is this correct?


    I've found the way to show Z(G) is a subgroup from here :

    http://en.wikipedia.org/wiki/Center_(group_theory)#As_a_subgroup


    So how does this differ from C(g), what other steps do I need?

    Thanks
     
  2. jcsd
  3. Feb 7, 2010 #2
    We say that two elements x, y of G commute if xy = yx. Suppose we have an element x of G, if it commutes with all other elements of G then it belongs to Z(G), which we call the center of G. Hence, Z(G) consists of all elements which commute with everything else. Notice that Z(G) depends only on G and doesn't refer to any specific element.

    On the other hand C(g) depends on a specific element g in G. It's the set of all elements of G which commutes with g.

    Try doing some examples.
     
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