Understanding Centralizers and Centers in Groups for Algebra Students"

  • Thread starter elle
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In summary, a centralizer in a group is a subgroup that consists of all elements that commute with a given element in the group. It is different from a center, which is a subgroup containing all elements that commute with every other element in the group. Understanding centralizers and centers is important in group theory as they provide information about the group's structure and properties. They can be calculated by finding elements that commute with a given element or every other element in the group. Centralizers and centers are unique for a given group, but may differ for different groups.
  • #1
elle
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Hi,
I'm currently self studying from an Algebra book but I'm having problems understanding the definition of the centralizer and center in a group :confused:

Like for example what is C(a) and Z(G) for the Quaternion group?

Can someone help? Many thanks!
 
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  • #2
Do you mean to say you're having trouble computing them, or understanding them?
 
  • #3
Both, would be very helpful if the definition was applied to the example excerise question I provided...
 

FAQ: Understanding Centralizers and Centers in Groups for Algebra Students"

What is a centralizer in a group?

A centralizer in a group is a subgroup that consists of all elements that commute with a given element in the group. In other words, the centralizer of an element is the set of elements that remain fixed when the given element is multiplied on either side.

How is a centralizer different from a center in a group?

A centralizer is a subgroup that contains a specific element, while a center is a subgroup that contains all elements in the group that commute with every other element. In other words, the centralizer of an element is a subset of the center of the group.

Why is understanding centralizers and centers important in group theory?

Centralizers and centers provide valuable information about the structure and properties of a group. They can help identify subgroups and determine whether a group is abelian or not. Additionally, understanding centralizers and centers can aid in solving problems related to group actions and symmetry.

How can centralizers and centers be calculated?

To calculate the centralizer of an element, you need to find all elements in the group that commute with that element. This can be done by multiplying the given element with every other element in the group and checking for commutativity. The center of a group can be calculated by finding all elements that commute with every other element in the group.

Are centralizers and centers unique for a given group?

Yes, centralizers and centers are unique for a given group. This means that for a specific group, there is only one centralizer and one center. However, for different groups, the centralizers and centers may differ.

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