2 edited tags
1

# Cohomology of $SL_2(\mathbb{F}_p)$ acting on trace zero matrices over $\mathbb{F}_p$

I've been reading a paper by R.Ramakrishna about lifting Galois representations and at some point he uses the fact that a representation with big image has trivial first cohomology group.

In other words, what i've been trying to prove (and failing at) is that given $p>5$ a prime, if $M$ is the group of trace zero $2\times 2$ matrices over $\mathbb{F}_p$ then:

$H^1(SL_2(\mathbb{F}_p),M) = 0$

where the action is given by conjugation.

My first approach was calculating the group $H^1(S_p,M)$, where $S_p$ is a $p$-sylow subgroup of $SL_2(\mathbb{F}_p)$. If it were trivial then the one that I want should be, because $M$ is a $p$-group, but it turns out it's not! (I think it's one dimensional).

Maybe there is some cohomological-theoretic reason I'm missing, thanks.