Let $\mathbb{Z}[Q_{32}]$ be the group ring where $Q_{32}=〈x,y|x^8=y^2,xyx=y〉$ is the quaternion group of order 32 and ε the augmentation map. We show that, if $PX=K(x-1)$ and $PX=K(-xy+1)$ has solution over $\mathbb{Z}[Q_{32}]$ and all $m\times m$ minors of $\epsilon (P)$ are relatively prime, then the linear system $PX=K$ has a solution over $\mathbb{Z}[Q_{32}]$. As a consequence of such results, we show that there is no map $f:W\to M_{Q_{32}}$ that is strongly surjective, i.e., such that $MR[f,a]=\min \{\mathrm{\#}(g^{-1}(a))|g\in [f]\}\ne 0$. Here, $M_{Q_{32}}$ is the orbit space of the 3-sphere ${\mathbb{S}}^3$ with respect to the action of $Q_{32}$ determined by the inclusion $Q_{32}\subseteq {\mathbb{S}}^3$ and W is a CW-complex of dimension 3 with $H^3(W;\mathbb{Z})=0$.

让 $\mathbb{ž}[{问}_{32}]$ 成为团体戒指 ${问}_{32}=〈X，ÿ|X^8={ÿ}^{\mathrm{2个}}，XÿX=ÿ〉$是阶数为32的四元数组，而ε是扩充图。我们证明，如果$PX=ķ（X-\mathrm{1个}）$ 和 $PX=ķ（-Xÿ+\mathrm{1个}）$ 解决了 $\mathbb{ž}[{问}_{32}]$ 和所有 $米\times 米$ 未成年人 $\epsilon （P）$ 相对质数，那么线性系统 $PX=ķ$ 有一个解决方案 $\mathbb{ž}[{问}_{32}]$。由于这种结果，我们表明没有地图$F：\mathrm{w\; ^}\to {\mathrm{中号}}_{{问}_{32}}$ 那是非常排斥的，即 $\mathrm{中号}\mathrm{[R}[F，\mathrm{一种}]=\mathrm{分}\{\mathrm{＃}（G^{-\mathrm{1个}}（\mathrm{一种}））|G\in [F]\}\ne 0$。这里，${\mathrm{中号}}_{{问}_{32}}$ 是3球的轨道空间 ${\mathbb{小号}}^3$ 关于...的行动 ${问}_{32}$ 由包含决定 ${问}_{32}\subseteq {\mathbb{小号}}^3$和w ^是一个CW -配合物尺寸3的与$H^3（\mathrm{w\; ^};\mathbb{ž}）=0$。