1. ## 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! :-)

2. 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,

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