On weak commutativity between $n$ isomorphic groups, $n \geq 2$
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.