Geodesic
2-Cohomologies of the groups $SL(n,q)$
Algebra i logika, Tome 47 (2008) no. 6, pp. 687-704

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Let $G=SL(n,q)$, where $q$ is odd, $V$ be a natural module over $G$, and $L=S^2(V)$ be its symmetric square. We construct a 2-cohomology group $H^2(G,L)$. The group is one-dimensional over $\mathbf F_q$ if $n=2$ and $q\neq3$, and also if $(n,q)=(4,3)$. In all other cases $H^2(G,L)=0$. Previously, such groups $H^2(G,L)$ were known for the cases where $n=2$ or $q=p$ is prime. We state that $H^2(G,L)$ are trivial for $n\ge3$ and $q=p^m$, $m\ge2$. In proofs, use is made of rather elementary (noncohomological) methods.
Keywords: cohomologies of groups, finite simple group. 
@article{AL_2008_47_6_a1,
     author = {V. P. Burichenko},
     title = {2-Cohomologies of the groups $SL(n,q)$},
     journal = {Algebra i logika},
     pages = {687--704},
     publisher = {mathdoc},
     volume = {47},
     number = {6},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2008_47_6_a1/}
}
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V. P. Burichenko. 2-Cohomologies of the groups $SL(n,q)$. Algebra i logika, Tome 47 (2008) no. 6, pp. 687-704. http://geodesic.mathdoc.fr/item/AL_2008_47_6_a1/