Center of a group vs centralizer vs conjugacy classes

Question

I am trying to fully understand the connection between a centralizer to conjugacy class.

Starting with the definitions:

Center: $Z(G)=\{z\in G:\forall g\in G, zg=gz\}$ which are all the elements in the group that commute

Centralizer: $C_G(S)=\{g\in G:gs=sg \text{ for all } s\in S\}$ which are all the elements of $g$ that commute with all of the elements subset of $S$

Conjugacy class: $Conj(x)=\{h\in G:\exists g\in G: gxg^{-1}=h \}=\{gxg^{-1}: g \in G \}$ can not explain which elements are in the conjugacy class

Center VS Centralizer: the center takes a group and returns all the elements in the group that commute whereas the Centralizer take a subset of the group and return only that elemnts of the subset which commute

So If for example all the subset commute , let say that the group is cyclic and there for every subset commute, we will have $Z(G)=C_G(G)$

Did I get that right? what are Conjugacy classes and how they differ from the centralizer and the center? is there a simple example for these definitions?

Answer 1

Saying that the center of $G$ are the elements of $G$ that commute makes no sense. Commute with what? The center of $G$ is the set of those elements of $G$ that commute with all other elements of $G$.

Your definition of centralizer is correct. Note that the centralizer of $G$ is precisely the center of $G$ (for every group). And that a group is commutative if and only if its center is the whole group.

As for the conjugacy classes: the conjugacy class of $x$ are all those elements of $G$ of the form $hxh^{-1}$. For instance, if $G$ is commutative, then the conjugacy class of each $x\in G$ is $x$ itself (because then $hxh^{-1}=hh^{-1}x=ex=x$). But if $G$ is the group $S_3$ (the group of the permutations of $\{1,2,3\}$), then you can check that the conjugacy class of $(1\ \ 2\ \ 3)$ is $\{(1\ \ 2\ \ 3),(1\ \ 3\ \ 2)\}$. Note that $x$ always belongs to the conjugacy class of $x$ (because $x=exe^{-1}$).

Answer 2

By definition You have $Z(G)=C_G(G)$. If an element is in the center, it commutes with all elements in $G$ as You correctly said and that means for $x\in Z(G)$ that $gxg^{-1}=x$ for all $g\in G$. The mappings $\phi_g:G\rightarrow G,x\mapsto gxg^{-1}$ are called the inner automorphisms of the group $G$ and make the group operate on itself. The conjugacy classes are the orbits of elements under this operation and the elements in the center are exactly those that have orbits of length $1$.