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I’m wondering what is known about the cohomology of $\operatorname{GL}_3(\mathbb{Z}_2)$, the general linear group of $3\times3$ matrices over the finite field $\mathbb{Z}_2$. There are results in Chapter I of Knudson’s “Homology of Linear Groups” as well as Chapter VII of Adem and Milgram’s “Cohomology of Finite Groups” that consider the cohomology of $\operatorname{GL}_n(\mathbb{F}_p)$, but not this specific case.

Update: It seems that page 178 of Adem and Milgram does the trick once we identify $\operatorname{GL}(3,2) =\operatorname{SL}(3,2)$. The relevant lemma says that $$H^*(\operatorname{SL}(3,2)) \oplus H^*(D_8) \cong H^*(S_4)\oplus H^*(S_4). $$

I’m wondering what is known about the cohomology of $\operatorname{GL}_3(\mathbb{Z}_2)$, the general linear group of $3\times3$ matrices over the finite field $\mathbb{Z}_2$. There are results in Chapter I of Knudson’s “Homology of Linear Groups” as well as Chapter VII of Adem and Milgram’s “Cohomology of Finite Groups” that consider the cohomology of $\operatorname{GL}_n(\mathbb{F}_p)$, but not this specific case.

Update: It seems that Adem and Milgram does the trick once we identify $\operatorname{GL}(3,2) =\operatorname{SL}(3,2)$. The relevant lemma says that $$H^*(\operatorname{SL}(3,2)) \oplus H^*(D_8) \cong H^*(S_4)\oplus H^*(S_4). $$

I’m wondering what is known about the cohomology of $\operatorname{GL}_3(\mathbb{Z}_2)$, the general linear group of $3\times3$ matrices over the finite field $\mathbb{Z}_2$. There are results in Chapter I of Knudson’s “Homology of Finite Groups” as well as Chapter VII of Adem and Milgram’s “Cohomology of Finite Groups” that consider the cohomology of $\operatorname{GL}_n(\mathbb{F}_p)$, but not this specific case.

Update: It seems that page 178 of Adem and Milgram does the trick once we identify $\operatorname{GL}(3,2) =\operatorname{SL}(3,2)$. The relevant lemma says that $$H^*(SL(3,2)) \oplus H^*(D_8) \cong H^*(S_4)\oplus H^*(S_4) $$

I’m wondering what is known about the cohomology of $\operatorname{GL}_3(\mathbb{Z}_2)$, the general linear group of $3\times3$ matrices over the finite field $\mathbb{Z}_2$. There are results in Chapter I of Knudson’s “Homology of Linear Groups” as well as Chapter VII of Adem and Milgram’s “Cohomology of Finite Groups” that consider the cohomology of $\operatorname{GL}_n(\mathbb{F}_p)$, but not this specific case.

Update: It seems that page 178 of Adem and Milgram does the trick once we identify $\operatorname{GL}(3,2) =\operatorname{SL}(3,2)$. The relevant lemma says that $$H^*(\operatorname{SL}(3,2)) \oplus H^*(D_8) \cong H^*(S_4)\oplus H^*(S_4). $$