8 clarity; more specific questioning

I stumbled across this question in a seminar-paper a long time ago:

Does there exist a positive integer $N$ such that if $G$ is a finite group with $\bigoplus_{i=1}^NH_i(G)=0$ then $G=\lbrace 1\rbrace$?

I believe this to still be an open problem. For $N=1$, any perfect group (ex: $A_5)$ A_5$) is a counterexample. For$N=2$, the binary icosahedral group$SL_2(\mathbb{F}_5)$SL_2(F_5)$ suffices (perfect group with periodic Tate cohomology). And I found in one of Milgram's papers a result for $N=5$, the sporadic Mathieu group $M23$. M_{23}$. Note that this question is answered for infinite groups, because we can always construct a topological space (hence a$BG$for some discrete group$G$) with prescribed homologies. Has any progress been made on this, or Is there another known group with a related questionlarger$N\ge 5$before homology becomes nontrivial?In particular, are Are there any classifications of obstructions in higher homology groups? []: Another view. A group is$\textit{acyclic}$if it has trivial integral homology. There are no nontrivial finite acyclic groups. Indeed, a result of Richard Swan says that a group with$p$-torsion has nontrivial mod-$p$cohomology in infinitely many dimensions, hence nontrivial integral homology. 7 added 28 characters in body I stumbled across this question in a seminar-paper a long time ago: Does there exist a positive integer$N$such that if$G$is a finite group with$\bigoplus_{i=1}^NH_i(G)=0$then$G=\lbrace 1\rbrace$? I believe this to still be an open problem. For$N=1$, any perfect group (ex:$A_5)$is a counterexample. For$N=2$, the binary icosahedral group$SL_2(\mathbb{F}_5)$suffices (perfect group with periodic Tate cohomology). And I found in one of Milgram's papers a result for$N=5$, the sporadic Mathieu group$M23$. Note that this question is answered for infinite groups, because we can always construct a topological space (hence a$BG$for some discrete group$G$) with prescribed homologies. Has any progress been made on this, or a related question? In particular, are there any classifications of obstructions in higher homology groups? []: Another way to viewthis question would be as follows. By the Kan-Thurston theorem, for any space$X$there exists a A group$G$with is$H_i(X)\cong H_i(G)$. Now \textit{acyclic}$ if $X$ is an it has trivial integral homology. There are no nontrivial finite acyclic spacegroups. Indeed, is a result of Richard Swan says that a group with $G$ necessarily infinite or trivial? An answer will solve the original questionp$-torsion has nontrivial mod-$p$cohomology in infinitely many dimensions, hence nontrivial integral homology. 6 deleted 170 characters in body I stumbled across this question in a seminar-paper a long time ago: Does there exist a positive integer$N$such that if$G$is a finite group with$\bigoplus_{i=1}^NH_i(G)=0$then$G=\lbrace 1\rbrace$? I believe this to still be an open problem. For$N=1$, any perfect group (ex:$A_5)$is a counterexample. For$N=2$, the binary icosahedral group$SL_2(\mathbb{F}_5)$suffices (perfect group with periodic Tate cohomology). And I found in one of Milgram's papers a result for$N=5$, the sporadic Mathieu group$M23$. Note that this question is answered for infinite groups, because we can always construct a topological space (hence a$BG$for some discrete group$G$) with prescribed homologies. Has any progress been made on this, or a related question? In particular, are there any classifications of obstructions in higher homology groups? []: Another way to view this question would be as follows. By the Kan-Thurston theorem, for any space$X$there exists a group$G$with$H_i(X)\cong H_i(G)$. Now if$X$is an acyclic space, is$G\$ necessarily infinite or trivial? A negative answer will solve the original question. Of course a positive An answer , however, will not solve the original questionbecause it will just imply that there exists no such space with the particular associated group.

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