It's useful for understanding the properties of certain groups. For example, if the center of a group is equivalent to the group itself, then the group is Abelian. Further, suppose you wanted to prove that a given dihedral group is of odd order; one way this can be done is by showing its center is equal to the trivial subgroup {e}.
Prof. Learnifyable: Thank you for making the videos on symmetries of an equilateral triangle and a square, and the center of a group. Will you be making a new video on the centralizer of a group soon? How about cyclic groups (long-awaited for)? Many thanks! > Benny Lo, Calif.
+Qaiser Abbas The center of a group G consists of the elements of G that commute with every element of G. There is also something called the centralizer. The centralizer of an element a in G is the set of all elements of G that commute with a. I hope that clears things up.
They are abstractions of modulo addition / multiplication; polygon transformation symmetries; and other wonky things. Previous videos in this series explicitly detail some table constructions, and there are many other tables.
For a dihedral group (rotations and symmetries of an n-gon) the group D_n = { (a^i)(b^j) | a^n = b^2 = e, ab = ba^(n-1) } we have the center Z( D_n ) = {e} , the trivial subgroup if n is odd and Z( D_n ) = {e, a^(n/2)} if n is even. Also as mentioned in the video, for any Abelian group Z(G) = G. There may be others that come from theorems for specific group structures.
The notation can vary, depending on which textbook you are using. Here is a video that sums up the notation that I use: th-cam.com/video/8cUiLMqNfHg/w-d-xo.html
Yes. Every element of the center of a group, Z(G), commutes with all elements of the group G. Then they will "mutually commute" as well, in the sense that elements of Z(G) commute with other elements of Z(G), because Z(G) is always a subset of (or equal to) G.
Great video! Thank you for taking the time to make this video.
+Kemal Tezer Dilsiz Thanks! I'm glad to hear that you liked it.
Helpful to understand the meaning of Z(G)
Good work, very well presented
Sir...ur videos r marvelous. Plzz try 2. Make some more videos on abstract algebra n...explain the concepts in depth......
It would be helpful to include some motivation for learning this. How is the center of a group useful?
it's useful for your soul LOL
To get a A in Abstract Algebra
It's useful for understanding the properties of certain groups. For example, if the center of a group is equivalent to the group itself, then the group is Abelian. Further, suppose you wanted to prove that a given dihedral group is of odd order; one way this can be done is by showing its center is equal to the trivial subgroup {e}.
to pass the exam duh!
Once you own properties subgroup, cyclic, and center, you can build houses and hotels on them and win Monopoly.
Prof. Learnifyable:
Thank you for making the videos on symmetries of an equilateral triangle and a square, and the center of a group. Will you be making a new video on the centralizer of a group soon? How about cyclic groups (long-awaited for)?
Many thanks!
> Benny Lo, Calif.
I'm planning a video on cyclic groups for next week. Stay tuned!
Hi how can I prove z(G) is subset of G
your videos are amazing!
Is it necessary to commute with all other elements of group to become centre of group or not....?
+Qaiser Abbas The center of a group G consists of the elements of G that commute with every element of G. There is also something called the centralizer. The centralizer of an element a in G is the set of all elements of G that commute with a. I hope that clears things up.
yes
Sir is it necessary for element to commute with all other elements of group to become centre of group please answer.
is'nt that the definition??? The elements of the subset Z(G) commute with every element of G.
Very nice sir
Thank u
Sir how to find group table?
They are abstractions of modulo addition / multiplication; polygon transformation symmetries; and other wonky things. Previous videos in this series explicitly detail some table constructions, and there are many other tables.
thanks a lot
why the word "Center?" Is there any kind of table or pictorial interpretation for selection of this word? Thank you
Centre is German for Zentrum
Can u tell me how to find a centre of a group without using cayley's table
For a dihedral group (rotations and symmetries of an n-gon) the group D_n = { (a^i)(b^j) | a^n = b^2 = e, ab = ba^(n-1) } we have the center Z( D_n ) = {e} , the trivial subgroup if n is odd and Z( D_n ) = {e, a^(n/2)} if n is even. Also as mentioned in the video, for any Abelian group Z(G) = G. There may be others that come from theorems for specific group structures.
Why is u2 not equal to the identity?
The notation can vary, depending on which textbook you are using. Here is a video that sums up the notation that I use:
th-cam.com/video/8cUiLMqNfHg/w-d-xo.html
thanks fya
is every elements of center of group mutualy cummute...?
Yes. Every element of the center of a group, Z(G), commutes with all elements of the group G. Then they will "mutually commute" as well, in the sense that elements of Z(G) commute with other elements of Z(G), because Z(G) is always a subset of (or equal to) G.
What is delta 1 and delta 2 in the D4 chart?
They refer to reflections of the square about the diagonal.
@@learnifyable ah ok
Sir ple pdf send kr dijiy
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