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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Originally Posted by Polyxendi 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, $\displaystyle \forall a,b \in Z(G), x \in G , (ab^{-1})x = a(b^{-1}x) = a(xb^{-1}) = (ax)b^{-1} = (xa)b^{-1} = x(ab^{-1})$
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