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I'm having trouble understanding the definition of "center" in group theory. My textbook says:

"The center, $Z(G)$, of a group is the subset of elements in $G$ that commute with every element of $G$. In symbols, $Z(G)= \{a \in G :\, ax= xa,\forall x\in G\}$."

What does it mean to commute with every element? Does this just mean it's an Abelian group or is it something entirely different?

This sounds pretty basic but I still don't understand.

Help!

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  • $\begingroup$ it means what you have written in your notation, by definition if for every $x\in G$ your candidate $a$ has the property that $ax=xa$ then $a$ is said to be in the center of the group. $\endgroup$ Feb 5, 2017 at 22:39
  • $\begingroup$ I think that in the groups $GL(n)$ the subgroup of diagonal matrices is a nice example of a center. $\endgroup$ Feb 6, 2017 at 21:15

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If the group $G$ is abelian, then we would have that $Z(G) = G$. Normally in a group which is not necessarily abelian, there are elements which do not commute with every other element. What is true, is that $$Z(G) \unlhd G$$ As an example that in general $Z(G) \neq G$, we have that $$Z(S_n) = \{\operatorname{id}\}$$ whenever $n \geq 3$ (a proof can be found here).

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I would use the quaternion group $$Q_8=\{\pm1,\pm i,\pm j,\pm k\}$$ as example.

The center of $Q_8$, $Z(Q_8)=\{\pm 1\}$.
This means that $1x=x1$ and $-1x=x(-1)$ is true when $x$ is any element of $Q_8$.
So $1$ and $-1$ commute with every element in $Q_8$.

In particular, $1$ and $-1$ commute with each other in $Z(Q_8)$.
This means that $Z(Q_8)$ is abelian.

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