For n≥6 the Kim-Manturov group Γ^4_n is finite, 2-step nilpotent, and has order 2^{binom(n,3)}.
Groups $\Gamma_n^4$: algebraic properties
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In the paper, groups $\Gamma_n^4$ closely connected with braid groups are researched from algebraic point of view. More exactly, for $n\geqslant7$, it is proved that $\Gamma_n^4$ is a nilpotent finite $2$-group with $4$-torsion and that its subgroup $(\Gamma_n^4)'$ is central.
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2025 1verdicts
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On the structure of groups defined by Kim and Manturov
For n≥6 the Kim-Manturov group Γ^4_n is finite, 2-step nilpotent, and has order 2^{binom(n,3)}.