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

[tgt]

conjugacy or a=g^-1bg occur a lot in algebra for a,b,g in G. But why?

 

[matt grime]

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?

 

[tgt]

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