How Do Centralizers and Generators Organize Group Elements?

[Locoism]

I'm having trouble grasping the concept of centralizers and generators. Is there any way to visualize these groups?

Edit:
For example, the centralizer of S in G (S is a subgp of G) is given by
Z(S) ={g ε G : gh = hg for all h ε G}

Does that just mean it is like the biggest abelian subgroup of G?
I'm pretty unclear on the concept of generator, and the notation use.

 

[lavinia]

I'm having trouble grasping the concept of centralizers and generators. Is there any way to visualize these groups?

Edit:
For example, the centralizer of S in G (S is a subgp of G) is given by
Z(S) ={g ε G : gh = hg for all h ε G}

Does that just mean it is like the biggest abelian subgroup of G?
I'm pretty unclear on the concept of generator, and the notation use.

the center of a group is the subgroup that commutes with every element of the group. The centralizer of a subgroup is the group that commutes with every element of the subgroup.


the centralizer does not need to be abelian. For instance the centralizer of the center is the entire group.

 

[Deveno]

let's pick a group, and see what we get for some different test values.

we'll pick S3, it's small, and non-abelian, so maybe we'll learn something.

now (1 2)(1 3) = (1 3 2), while (1 3)(1 2) = (1 2 3), so neither (1 2) or (1 3)

can be in the center of S3.

(1 2)(1 2 3) = (2 3), while (1 2 3)(1 2) = (1 3), so (1 2 3) isn't in the center.

(1 2)(1 3 2) = (1 3), (1 3 2)(1 2) = ( 2 3), so (1 3 2) isn't in the center.

(1 2)(2 3) = (1 2 3), (2 3)(1 2) = (1 3 2), so (2 3) isn't in the center.

so Z(S3) is just the identity (S3 is VERY non-abelian, hardly anything commutes).

let's see what happens if we try to centralize a smaller set.

let's choose H = {1, (1 2 3), (1 3 2)}.

straight-away we see that (1 2), (1, 3) and (2, 3) aren't in Z(H), from our investigations into the center.

but (1 2 3)^-1 = (1 3 2), and everything commutes with its inverse, so

Z(H) = H.

note that if we pick S = {1}, everything commutes with the identity, so Z({1}) = S3.

notice that the smaller S gets, the bigger Z(S) got. Z(S) is sort of a way of telling:

"how abelian is S compared to the rest of G".

the identity subgroup is VERY abelian, so it makes sense that Z({1}) is big. S3 is not very abelian, so it makes sense Z(S3) is small. {1,(1 2 3),(1 3 2)} is sort of "in the middle", everything in it commutes with itself (because it's an abelian subgroup), but it doesn't commute with anything outside of it.

if a subgroup H is abelian, Z(H) will contain all of H, and maybe more.

if a subgroup H is not abelian, Z(H) won't contain all of H.

if the main group G is abelian, of course, Z(H) = G for every subgroup H.

the center of G, Z(G) will always be an abelian group (since everything in it commutes with everything, including its own elements), but it isn't necessarily the largest abelian subgroup of G. for example, in the group D4 =

{1,r,r^2,r^3,s,rs,r^2s,r^3s}, the center is {1,r^2}, but the subgroup {1,r,r^2,r^3} is abelian and is clearly larger.

 

[Locoism]

Thank you Deveno that was really helpful. Man this stuff is abstract...

 

[blue_raver22]

Centralizers and generators are important concepts in group theory that help us understand the structure and behavior of groups. To visualize these groups, it may be helpful to think of them as a way to organize and categorize elements within a group.

The centralizer of an element S in a group G is the set of all elements in G that commute with S. In other words, for any element h in the centralizer, the operation of multiplying h by S and multiplying S by h will result in the same element. This can be thought of as a "center" of sorts within the group, where all elements in the centralizer are "connected" to S.

As for the example given, Z(S) represents the centralizer of S in G, and it is indeed the largest abelian subgroup of G. This is because all elements in the centralizer commute with each other, making it an abelian subgroup.

On the other hand, generators refer to elements that can generate the entire group. In other words, by combining the generator with itself and with other elements in the group, we can create all other elements in the group. This can be visualized as a "starting point" or "building block" for the group.

The notation used for generators can vary, but a common one is <g>, which represents the subgroup generated by g. This means that all elements in the subgroup can be created by combining g with itself and with other elements in the group.

Overall, centralizers and generators are important tools in understanding the structure and behavior of groups. They allow us to categorize and organize elements within a group, and can help us analyze the properties of the group as a whole. I hope this helps clarify the concepts for you.