Conjugacy in Algebra: Why Does a=g^-1bg Occur?

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In summary, conjugacy or a=g^-1bg is a common concept in algebra, where a, b, and g are elements in a group G. This is because conjugate elements share similar properties due to the isomorphism of the map f_g. The set of all such maps is known as the group of inner automorphisms. In linear algebra, conjugate matrices also have many shared properties. Additionally, the number of conjugacy classes in a finite group is equivalent to the number of simple complex valued representations. Overall, while they may seem strange at first, conjugate elements play an important role in understanding various mathematical concepts.
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tgt
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conjugacy or a=g^-1bg occur a lot in algebra for a,b,g in G. But why?
 
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Because it's important?

Conjugate elements (in a group) have the 'same' properties, essentially. This is because the map

f_g :G-->G

f_g(x)=g^{-1}xg

is an isomorphism. The set of all such f_g, g in G is the group of inner automorphisms. In a lot of cases these are all automorphisms of a finite group; in some cases they are not.

In linear algebra, conjugate matrices share many properties...

The number of conjugacy classes of a finite group is the same as the number of simple complex valued representations.

Shall I go on?
 
  • #3
They first seem a bit weird but now that you mentioned these things, they seem quiet natural.
 

1. What is conjugacy in algebra?

Conjugacy in algebra refers to the relationship between elements in a group that are related by a certain operation. In particular, it describes the relationship between an element and its inverse.

2. Why is conjugacy important in algebra?

Conjugacy is important in algebra because it helps us understand the structure and properties of groups. It allows us to simplify complex expressions and make connections between different elements in a group.

3. What does the expression a=g^-1bg mean?

This expression represents conjugacy in algebra, where a and b are elements in a group and g is a fixed element. It means that a and b are conjugate to each other under the operation of g.

4. How does conjugacy relate to symmetry?

Conjugacy is closely related to symmetry because it describes how different elements in a group can be transformed into each other using a certain operation. This is similar to how symmetrical objects can be transformed into each other through reflections, rotations, or translations.

5. Can conjugacy occur in other mathematical structures besides groups?

Yes, conjugacy can also occur in other mathematical structures, such as rings and fields. In these structures, the concept of conjugacy is used to describe the relationship between elements under certain operations, similar to how it is used in groups.

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