Zulip Chat Archive

Stream: maths

Topic: group acting on group


view this post on Zulip Kenny Lau (Apr 09 2018 at 13:05):

Do we have a name for a group acting on another group that is compatible with the group structure?

view this post on Zulip jmc (Apr 09 2018 at 13:19):

There is this thing called G-module

view this post on Zulip jmc (Apr 09 2018 at 13:19):

But then you act on abelian groups

view this post on Zulip Kenny Lau (Apr 09 2018 at 13:19):

but a group is not a module

view this post on Zulip Kenny Lau (Apr 09 2018 at 13:19):

right

view this post on Zulip jmc (Apr 09 2018 at 13:21):

Where does this show up?

view this post on Zulip Kenny Lau (Apr 09 2018 at 13:21):

in my brain

view this post on Zulip Kenny Lau (Apr 09 2018 at 13:21):

i'm just making this up

view this post on Zulip Kenny Lau (Apr 09 2018 at 13:22):

oh wait, this does show up in group theory

view this post on Zulip Kenny Lau (Apr 09 2018 at 13:22):

a group acts on a normal subgroup by conjugation

view this post on Zulip jmc (Apr 09 2018 at 13:22):

Ok, G-modules show up a lot in group cohomology

view this post on Zulip jmc (Apr 09 2018 at 13:22):

Aah, ok, sure

view this post on Zulip Kenny Lau (Apr 09 2018 at 13:22):

right, i'm learning group cohomology right now

view this post on Zulip Kenny Lau (Apr 09 2018 at 13:22):

and then an action of G on N is just a homomorphism G -> Aut(N)

view this post on Zulip jmc (Apr 09 2018 at 13:23):

Sure, but if G acts on G', it is also just a homom G -> Aut(G')

view this post on Zulip jmc (Apr 09 2018 at 13:23):

if the action is compatible with the group structure

view this post on Zulip Kevin Buzzard (Apr 09 2018 at 13:27):

If the group M you're acting on is abelian, then this is just called a G-module usually (G the group doing the acting)

view this post on Zulip Kevin Buzzard (Apr 09 2018 at 13:28):

If M is not abelian then this is sometimes called a "non-abelian G-module" and the theory very quickly gets technical

view this post on Zulip Kevin Buzzard (Apr 09 2018 at 13:28):

H^1 is no longer a group, but just a pointed set

view this post on Zulip Kevin Buzzard (Apr 09 2018 at 13:28):

and for higher cohomology groups one has to use fancy stuff like gerbes

view this post on Zulip Kevin Buzzard (Apr 09 2018 at 13:29):

See Serre's book on Galois cohomology (some appendix) for a brief and clear introduction.


Last updated: May 12 2021 at 07:17 UTC