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Math Help - Abtract algebra help!

  1. #1
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    Abtract algebra help!

    For any group G, the set of all the elements that commute with every element of G is called the center of G and it's denoted by:

    Z(G) = {a ∈ G, such that ax = xa, ∀ x ∈ G}

    Prove that Z(G) is a subset of G.


    Any ideas?

    Thanks! :-)
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  2. #2
    Lord of certain Rings
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    Quote Originally Posted by Polyxendi View Post
    For any group G, the set of all the elements that commute with every element of G is called the center of G and it's denoted by:

    Z(G) = {a ∈ G, such that ax = xa, ∀ x ∈ G}

    Prove that Z(G) is a subset of G.


    Any ideas?

    Thanks! :-)
    I think you mean:
    "Prove that Z(G) is a subgroup of G"

    In that case,

    \forall a,b \in Z(G), x \in G , (ab^{-1})x = a(b^{-1}x) = a(xb^{-1}) = (ax)b^{-1} = (xa)b^{-1} = x(ab^{-1})
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