On weak commutativity between n isomorphic groups, n ≥ 2
Abstract
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 Hψ 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 χn (−) also preserves the same classes. However, we observe that for n ≥ 2 many of the 2n H-generating copies of χn (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, Hψ0 = H, Hψ1 = Hψ). We study quotients of χn (H) which are also simultaneously quotients of χ2n (H); we refer to them as Hybrid χn (H) - χ2n (H) groups. For n=2 we determine the largest common quotient Q(H) of χ2(H) and χ4(H), that is, the largest hybrid χ2(H)-χ4(H) group. A related class of groups in our study is χ(n,H), a quotient of χn(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, χn(H) behaves differently: χ3(C2 × C2) is infinite, while χ3(Cpt × Cpu) is finite for p odd.
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BibTeX
@misc{sidki2026weak,
title = {{On weak commutativity between $n$ isomorphic groups, $n \geq 2$}},
author = {Said N. Sidki and Noraí R. Rocco and Ricardo N. de Oliveira and Dmytro Savchuk},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07330},
eprint = {2610.07330},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07330},
}
RIS
TY - UNPB
TI - On weak commutativity between n isomorphic groups, n ≥ 2
AU - Said N. Sidki
AU - Noraí R. Rocco
AU - Ricardo N. de Oliveira
AU - Dmytro Savchuk
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.07330
UR - https://arxiv.org/abs/2610.07330
AB - 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 ≥ 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 × C_2) is infinite, while χ_3(C_p^t × C_p^u) is finite for p odd.
ER -