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Mathematical analysis demonstrates that subgroup nilpotency and Sylow towers in finite groups can be identified via commutator subsets, indicating robust structural detection criteria.
Given a finite group 𝐺 and an element 𝑥 of 𝐺, let K x ( G ) Kₓ(G) denote the set of commutators [ x , g ] [x,g] as 𝑔 varies through 𝐺. We prove that various group theoretical properties of 𝐺, for instance, nilpotency of several subgroups of 𝐺 and the existence of a Sylow tower for 𝐺, can be detected by considering various subsets of 𝐺 of the form K x ( H ) Kₓ(H) where H ⩽ G H G and x ∈ H x∈ H .
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Souza et al. (2026) studied this question.
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