发表机构
Universidade de Brasília; Universidade Federal de Goiás（巴西利亚大学; 戈亚斯联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对 $n$ 个同构群之间的弱交换性，引入 $n$ 元弱交换群 $\chi_n(H)$，研究其与迭代算子 $\chi^n(H)$ 的混合商群，确定 $n=2$ 时的最大公共商，并证明相关群类保持有限性、幂零性和完全性。
AI 中文摘要
弱交换群由第一作者在20世纪80年代初定义为 $\chi(H) = \langle H, H^\psi \mid [h,h^\psi] = 1, \\ \forall h \in H \rangle$，其中 $H$ 和 $H^\psi$ 在同构 $\psi$ 下是同构群，并证明了 $\chi(-)$ 作为群算子保持了可解群、有限群、有限 $p$-群和完全群这些类。自那以后，这个列表已大幅增长（见参考文献）。显然，迭代算子 $\chi^n (-)$ 也保持相同的类。然而，我们观察到对于 $n \geq 2$，$\chi^n (H)$ 的 $2^n$ 个由 $H$ 生成的副本中，许多副本彼此之间不能弱交换。我们在本文中引入了由 $n$ 个彼此弱交换的 $H$ 副本生成的“$n$ 元弱交换群”，即 $\chi_n (H) = \langle H, H^\psi, \dots, H^{\psi_{n-1}} \mid [h^{\psi_i}, h^{\psi_j}] = 1, \\ \forall h \in H, \\ 0 \leq i < j \leq n-1 \rangle$（按定义，$H^{\psi_0} = H, H^{\psi_1} = H^\psi$）。我们研究 $\chi^n (H)$ 的商群，这些商群同时也是 $\chi_{2^n} (H)$ 的商群；我们称它们为混合 $\chi^n (H)$-$\chi_{2^n} (H)$ 群。对于 $n=2$，我们确定了 $\chi^2(H)$ 和 $\chi_4(H)$ 的最大公共商 $Q(H)$，即最大的混合 $\chi^2(H)$-$\chi_4(H)$ 群。我们研究中的一类相关群是 $\chi(n,H)$，它是 $\chi_n(H)$ 的商群，由第三作者在其2007年的博士论文中先前引入。我们证明这类群继续保持有限性、幂零性和完全性，并提供关于其结构的详细信息。相比之下，$\chi_n(H)$ 的行为不同：$\chi_3(C_2 \times C_2)$ 是无限的，而 $\chi_3(C_{p^t} \times C_{p^u})$ 对于奇素数 $p$ 是有限的。
英文摘要
The Weak Commutativity Group was defined by the first author in the early 1980's as $ χ(H) = \langle H, H^ψ\mid [h,h^ψ] = 1, \ \forall h \in H \rangle $ where $H$ and $H^ψ$ are isomorphic groups under an isomorphism $ψ$, and proved that $χ(-)$, seen as a group operator, preserved the classes of solvable groups, finite groups, finite $p$-groups, and perfect groups. The list has grown considerably since then (see references). It is clear that the iterated operator $χ^n (-)$ also preserves the same classes. However, we observe that for $n \geq 2$ many of the $2^n$ $H$-generating copies of $χ^n (H)$ fail to weakly commute among themselves. We introduce in this paper the "$n$-ary Weak Commutativity Group" generated by $n$ copies of $H$ which weakly commute among themselves; that is $ χ_n (H) = \langle H, H^ψ, \dots, H^{ψ_{n-1}} \mid [h^{ψ_i}, h^{ψ_j}] = 1, \ \forall h \in H, \ 0 \leq i < j \leq n-1 \rangle $ (by definition, $H^{ψ_0} = H, H^{ψ_1} = H^ψ$). We study quotients of $χ^n (H)$ which are also simultaneously quotients of $χ_{2^n} (H)$; we refer to them as Hybrid $χ^n (H)$ - $χ_{2^n} (H)$ groups. For $n=2$ we determine the largest common quotient $Q(H)$ of $χ^2(H)$ and $χ_4(H)$, that is, the largest hybrid $χ^2(H)$-$χ_4(H)$ group. A related class of groups in our study is $χ(n,H)$, a quotient of $χ_n(H)$, previously introduced by the third named author in his 2007 doctoral thesis. We prove that this class of groups continues to preserve finiteness, nilpotency and perfectness, and we provide detailed information about their structure. In contrast, $χ_n(H)$ behaves differently: $χ_3(C_2 \times C_2)$ is infinite, while $χ_3(C_{p^t} \times C_{p^u})$ is finite for $p$ odd.
Comments32 pages. With an appendix by Dmytro Savchuk
