Centre and centraliser, concept

Question

Let G be a group.$Z(G )$ is the centre of the group $G$, i.e $Z(G)$ contains all the elements of the group $G$ which commutes with every element of the group. And $C(a)$ is the centraliser i.e $C(a)$ contains those elements which commutes only with a.

Now my question is that, it seems to be $Z(G)$ is a bigger subgroup than $C(a)$.And also I think $C(a) \subset Z(G)$,but all the books are saying its opposite, why?

And I also notice that $Z(G)=\bigcap_{a\in G}C(a)$,but, I think its oppsite notion i.e $C(a) = \bigcap_{a\in G}Z(G)$. Please clear my concept...

Answer 1

Suppose that $a,b,c\in G$ and $b$ commutes with $a$ but $b$ does not commute with $c$, then $b\in C(a)$ but $b\notin Z(G)$. On the other hand if $b\in Z(G)$ then $b$ is certainly in $C(a)$.

Answer 2

Now my question is that, it seems to be Z(G) is a bigger subgroup than C(a).

Commuting with everything is a more restrictive condition than commuting with a single thing. Therefore there should be less elements that satisfy the stricter condition.

To give a different example: the condition of being at least as tall as Shampa Das is less restrictive than being at least as tall as all humans, therefore we should expect the people (in theory, a single person) who are as at least as tall as all humans to be a subset of the people who are at least as tall as Shampa Das.

And also I think C(a) \subset Z(G), but all the books are saying its opposite,why?

It's easy to think of a counterexample. For instance, take $a$ not in the center of $G$. Obviously $a$ commutes with itself, so $a\in C(a)$ but not in $Z(G)$, so obviously the containment you describe isn't true.

And I also notice that $Z(G)=\bigcap_{a\in G}C(a)$,but, I think its oppsite notion i.e $C(a) = \bigcap_{a\in G}Z(G)$.

Well, let's consider the second thing you wrote: you are taking the intersection of $Z(G)$ with itself lots of times, it doesn't depend on $a$ at all in any way. Therefore $\bigcap_{a\in G}Z(G)=Z(G)$, and it seems you already know this is not often equal to $C(a)$. So that expression clearly has problems.

In the former description $Z(G)=\bigcap_{a\in G}C(a)$, what do you have: You are taking the things which commute with $a$ for each $a$, and then seeing what is common to all of them: in other words, you're finding out what elements commute with every element of $G$. That is precisely the center of $G$!