For odd primes p, the E2 page of the LHS spectral sequence for the Sylow p-subgroups of GL_2(Z/p^n) and SL_2(Z/p^n) is independent of n>1, with stable elements described via fusion systems.
On the modular cohomology of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$
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abstract
Let $p$ be an odd prime. Denote a Sylow $p$-subgroup of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$ by $S_p(n,GL)$ and $S_p(n,SL)$ respectively. The theory of stable elements tells us that the mod-$p$ cohomology of a finite group is given by the stable elements of the mod-$p$ cohomology of it's Sylow $p$-subgroup. We prove that for suitable group extensions of $S_p(n,GL)$ and $S_p(n,SL)$ the $E_2$-page of the Lyndon-Hochschild-Serre spectral sequence associated to these extensions does not depend on $n>1$. Finally, we use the theory of fusion systems to describe the ring of stable elements.
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On the modular cohomology of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$
For odd primes p, the E2 page of the LHS spectral sequence for the Sylow p-subgroups of GL_2(Z/p^n) and SL_2(Z/p^n) is independent of n>1, with stable elements described via fusion systems.