We compute the equivariant $KO$-homology of classifying space for proper actions $\textrm{SL}_3(\mathbb{Z})$ and $\textrm{GL}_3(\mathbb{Z})$. also Bredon homology $K$-homology spaces $\textrm{PSL}_2(\mathbb{Z}[\frac{1}{p}])$ $\textrm{SL}_2(\mathbb{Z}[\frac{1}{p}])$ each prime $p$. Finally, we prove Unstable Gromov-Lawson-Rosenberg Conjecture a large class groups whose maximal finite subgroups a...

‎given an integer $n$‎, ‎we denote by $mathfrak b_n$ and $mathfrak c_n$ the classes of all groups $g$ for which the map $phi_{n}:gmapsto g^n$ is a monomorphism and an epimorphism of $g$‎, ‎respectively‎. ‎in this paper we give a characterization for groups in $mathfrak b_n$ and for groups in $mathfrak c_n$‎. ‎we also obtain an arithmetic description of the set of all integers $n$ such that a gr...

let $g={rm sl}_2(p^f)$ be a special linear group and $p$ be a sylow‎ ‎$2$-subgroup of $g$‎, ‎where $p$ is a prime and $f$ is a positive‎ ‎integer such that $p^f>3$‎. ‎by $n_g(p)$ we denote the normalizer of‎ ‎$p$ in $g$‎. ‎in this paper‎, ‎we show that $n_g(p)$ is nilpotent (or‎ ‎$2$-nilpotent‎, ‎or supersolvable) if and only if $p^{2f}equiv‎ ‎1,({rm mod},16)$‎.