Is it possible to find an isomorphism between two groups with a new operation?

[TorKjellsson]

Homework Statement

Let (G,\cdot) be a group. Defining the new operation * such that a*b = b \cdot a it is pretty easy to show that (G,*) is a group. Show that this new group is isomorphic to the old one.

Homework Equations

The Attempt at a Solution

I have been experimenting with the possibility to define an isomorphism \phi (a*b) = b a but can't really seem to get it right. Can anyone give me a hint on how to find the isomorphism?


Tor

 

[Oster]

Will the conjugate action work?

 

[TorKjellsson]

Hm, actually after doing some more exercises I figured out a function that works. I am not quite sure of what you mean by conjugate action but I post my solution here:

Let \phi: (G,*) \rightarrow (G, \cdot) such that \phi (a) = a^{-1}. Then \phi(a) \phi(b) = a^{-1} b^{-1}. Consider now \phi(a*b) = (ba)^{-1} = a^{-1} b^{-1} and we see that \phi(a) \phi(b) = \phi(a*b).

 

[Oster]

Ooo nice. I was thinking of something like a --> ga(g-1). But that doesn't work.

 

[Kreizhn]

This group is called the opposite group. It is important for studying the relation between left- and right- group actions. Just thought I'd throw that fun fact in there.