# regular representations

• Sep 1st 2009, 03:34 AM
alexandrabel90
regular representations
Can someone explain what does right and left regular representations of a group G mean and the purpose of it?

thanks!
• Sep 1st 2009, 03:57 AM
ynj
Quote:

Originally Posted by alexandrabel90
Can someone explain what does right and left regular representations of a group G mean and the purpose of it?

thanks!

Left regular representation means a map $\displaystyle \lambda_a:G\rightarrow G,\lambda_a(g)=ag$.
• Sep 1st 2009, 05:03 AM
alexandrabel90
thank you!

but when is the left or right regular representation used?
• Sep 1st 2009, 05:04 AM
ynj
Quote:

Originally Posted by alexandrabel90
thank you!

but when is the left or right regular representation used?

To prove that every group is isomorphic to a permutation group.
• Sep 1st 2009, 05:12 AM
alexandrabel90
sorry, im new to this topic so im rather lost. Those that mean that is there is a left and right regular representation for a group G, it is isomorphic? or what?
• Sep 1st 2009, 05:39 AM
ynj
Quote:

Originally Posted by alexandrabel90
sorry, im new to this topic so im rather lost. Those that mean that is there is a left and right regular representation for a group G, it is isomorphic? or what?

no,it actually means that each $\displaystyle \lambda_a$ is a permutation on $\displaystyle G$, and all these $\displaystyle \lambda_a$form a group which is isomorphic to $\displaystyle G$.
All the left regular representations of G form a group that is isomorphic to G, so does the right regular representations.