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

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SUMMARY

Conjugacy in algebra, represented by the equation a = g-1bg, is a fundamental concept where elements a, b, and g belong to a group G. This relationship arises because conjugate elements share essential properties, facilitated by the isomorphism defined by the map fg: G → G, where fg(x) = g-1xg. The set of all such maps forms the group of inner automorphisms, which can be automorphisms of finite groups. Additionally, the number of conjugacy classes in a finite group corresponds to the number of simple complex-valued representations.

PREREQUISITES
  • Understanding of group theory concepts, specifically conjugacy
  • Familiarity with isomorphisms and automorphisms in algebra
  • Knowledge of linear algebra, particularly properties of conjugate matrices
  • Basic comprehension of representation theory in finite groups
NEXT STEPS
  • Study the properties of inner automorphisms in group theory
  • Explore the relationship between conjugacy classes and representations in finite groups
  • Learn about isomorphisms in algebraic structures
  • Investigate the significance of conjugate matrices in linear algebra
USEFUL FOR

Mathematicians, algebra students, and researchers interested in group theory, representation theory, and linear algebra will benefit from this discussion on conjugacy.

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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?
 
They first seem a bit weird but now that you mentioned these things, they seem quiet natural.
 

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