# Questioning Centralizers: A Group Acting On Itself

• 0rthodontist
In summary, The conversation is about a problem involving a group acting on itself by conjugation. The question asks to show that if two elements, a and b, are conjugates in the group, then their centralizers are equal if and only if they are normal subgroups of the group. The person speaking mentions having difficulty with one direction of the proof and the professor clarifies that the question should have stated that a and b are distinct. The professor also mentions discussing the problem in class and finding a counterexample in S6.
0rthodontist
This assignment has not yet been turned in, so I do not want any unfair help on this whatsoever. Please just give me a "yes" or a "no." Do not explain.

The question:
Let G be a group acting on itself by conjugation. Show that if a and b are conjugates in G, then the centralizers C(a) and C(b) are equal iff these centralizers are normal subgroups of G.

My problem:
I got the normal -> equal direction. But a is a conjugate of itself, and C(a) = C(a), so the other direction would imply that any centralizer is normal. I may be so tired I'm blind, but is this a flawed question?

I've handed the homework in. The professor said that the problem should have said that a and b are distinct.

The professor talked about this problem today: he had found a counterexample in S6 to the revised proposition stipulating that a and b are distinct. Apparently the other direction should have read something like "if a is in G, and for every b that is a conjugate of a, C(a) = C(b), then C(a) is normal in G."

## 1. What is the purpose of studying "Questioning Centralizers: A Group Acting On Itself"?

The purpose of studying "Questioning Centralizers: A Group Acting On Itself" is to understand the concept of centralizers in group theory and how a group can act on itself through questioning. This topic has applications in various fields such as mathematics, computer science, and physics. It also helps in understanding the structure and properties of groups and their elements.

## 2. Can you explain the concept of centralizers in group theory?

Centralizers in group theory refer to the subgroup of a group that consists of all the elements that commute with a specific element of the group. In other words, it is the set of elements that remain unchanged when multiplied by a given element. It is denoted by Z(g) where g is the element of the group.

## 3. How does a group act on itself through questioning?

A group can act on itself through questioning by using a specific element of the group to ask questions to other elements. This element, also known as the centralizer, remains unchanged when multiplied by the other elements, and hence the group acts on itself through questioning. This concept is used to study the structure and properties of groups and their elements.

## 4. What are some real-world applications of "Questioning Centralizers: A Group Acting On Itself"?

The concept of "Questioning Centralizers: A Group Acting On Itself" has various real-world applications. In computer science, it is used in cryptography to study the structure of codes and their security. In physics, it helps in understanding the symmetries of physical systems. It also has applications in chemistry, sociology, and economics.

## 5. Is there any significance of studying "Questioning Centralizers: A Group Acting On Itself" in mathematics?

Yes, studying "Questioning Centralizers: A Group Acting On Itself" is significant in mathematics. It helps in understanding the structure and properties of groups, which are essential in various branches of mathematics such as algebra, geometry, and number theory. It also has applications in other fields of mathematics, such as topology and differential equations.

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