group acting on group

Stream: maths

[Kenny Lau]

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

[jmc]

There is this thing called G-module

[jmc]

But then you act on abelian groups

[Kenny Lau]

but a group is not a module

[Kenny Lau]

right

[jmc]

Where does this show up?

[Kenny Lau]

in my brain

[Kenny Lau]

i'm just making this up

[Kenny Lau]

oh wait, this does show up in group theory

[Kenny Lau]

a group acts on a normal subgroup by conjugation

[jmc]

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

[jmc]

Aah, ok, sure

[Kenny Lau]

right, i'm learning group cohomology right now

[Kenny Lau]

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

[jmc]

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

[jmc]

if the action is compatible with the group structure

[Kevin Buzzard]

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)

[Kevin Buzzard]

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

[Kevin Buzzard]

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

[Kevin Buzzard]

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

[Kevin Buzzard]

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