Center of Group: Great Thanks!

In summary, the conversation discusses finding a value z that satisfies zg=gz for all g in G, where z is defined as a matrix. It is shown that for certain choices of g, z must have specific properties, leading to the conclusion that z must be a nonzero scalar matrix. The poster also expresses frustration with people erasing their posts after they have been answered, hindering the learning process for others.
  • #1
Monkeyfry180
7
0
Sounds great thank you
 
Last edited:
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  • #2
The 2x2 case is easy (for the general case, see e.g. www-math.mit.edu/~dav/genlin.pdf).

You want to find z such that zg=gz for all g in G.
Let z = {{z11,z12},{z21,z22}}
If you choose g = {{0,1},{1,0}} then you find that
z11 = z22 and z12 = z21
If you choose g = {{0,1},{-1,0}} then you get
z11 = z22 and z12 = -z21
Thus z12 = z21 = 0 and z must be a nonzero (real) http://en.wikipedia.org/wiki/Scalar_matrix" .
 
Last edited by a moderator:
  • #3
I would appreciate it if people would NOT erase their post after it has been answered. Other people use these posts to learn and erasing the original question makes that very difficult.
 
  • #4
@HallsofIvy.
I was just thinking the same thing. Maybe if there was an option to look a the history of an edited post then it wouldn't be so much of an issue?
 
  • #5
I am always excited to hear positive feedback from members of the community. I am glad to hear that our work has been helpful to you and we appreciate your support. Thank you for your kind words and we will continue to strive for excellence in our research and contributions to the scientific community.
 

What is the "Center of Group"?

The "Center of Group" is a term used in group theory to refer to the element that is the average of all the elements in a group. It is also known as the group's centroid or barycenter.

Why is the "Center of Group" important?

The "Center of Group" is important because it helps to characterize the structure and behavior of a group. It can be used to determine if a group is symmetric or asymmetric, and can also provide information about the group's subgroups and substructures.

How is the "Center of Group" calculated?

The "Center of Group" is typically calculated by taking the sum of all the elements in a group and dividing by the number of elements. In some cases, it may also involve taking the average of specific elements or using other mathematical operations.

Can the "Center of Group" be found in any type of group?

Yes, the "Center of Group" can be found in any type of group, whether it is a finite or infinite group, a cyclic or non-cyclic group, or a commutative or non-commutative group. However, the method for calculating it may vary depending on the type of group.

What is the significance of the phrase "Great Thanks!" in relation to the "Center of Group"?

The phrase "Great Thanks!" is often used in the context of group theory to express gratitude for the usefulness and versatility of the concept of the "Center of Group". It acknowledges the important role that the "Center of Group" plays in understanding and analyzing groups.

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