Groups of nilpotency class $ 2 $ and exponent $ p $ (with $ p $ odd) admit concrete coordinate models governed by alternating $ \mathbb{F}_p $-bilinear commutator data, a viewpoint central in both the structure theory of $ p $-groups and algorithmic approaches to isomorphism testing and related problems. Motivated by decomposition primitives in computational group theory and by rank-based filters used in modern isomorphism pipelines, we introduce the commutator-constrained factorization problem: given $ g\in G $ and a target commutator value $ h\in [G, G] $, count and construct pairs $ (x, y) $ with $ xy = g $ and $ \left[x, y\right] = h $. In an explicit bilinear-data input model, we show that the decision, counting, and search variants reduce to solvability of a single linear system $ T_w(u) = h $ over $ \mathbb{F}_p $, where $ T_w $: $ u\mapsto b(u, w) $ is the contraction map determined by the $ V $-projection $ w $ of $ g $. When solvable, the solution set $ \mathcal{S}(g, h) $ is exhibited as an explicit torsor for $ \ker(T_w)\times W $, yielding a closed counting formula and a certified witness construction by elementary linear algebra. We derive exact secondary laws: a complete description of the attainable commutator set $ \mathcal{H}(g) = \operatorname{im}(T_w) $, an exact uniformity law over attainable values, and a factor-swap bijection relating $ \mathcal{S}(g, h) $ to a shifted product constraint. Finally, we define rank-profile polynomials $ P_G(t) = \sum_{w\in V} t^{\operatorname{rank}(T_w)} $, prove isomorphism invariance, and extract further invariants (radical size, extremal attainable-set size) directly from counting oracles. The odd-prime hypothesis is made explicit throughout: the centered coordinate law uses the scalar $ 1/2\in\mathbb F_p $, whereas characteristic two requires a cocycle or quadratic-refinement formulation. We also record coordinate-invariance, conversion costs from power-commutator input, sparse implementation refinements, and limitations of the rank-profile invariant.
Citation: Ghaliah Alhamzi, Mdi Begum Jeelani, Wael Mahmoud Mohammad Salameh, Prakash Jadhav. Commutator-constrained factorizations in class-two exponent-p-groups[J]. AIMS Mathematics, 2026, 11(6): 15561-15580. doi: 10.3934/math.2026640
Groups of nilpotency class $ 2 $ and exponent $ p $ (with $ p $ odd) admit concrete coordinate models governed by alternating $ \mathbb{F}_p $-bilinear commutator data, a viewpoint central in both the structure theory of $ p $-groups and algorithmic approaches to isomorphism testing and related problems. Motivated by decomposition primitives in computational group theory and by rank-based filters used in modern isomorphism pipelines, we introduce the commutator-constrained factorization problem: given $ g\in G $ and a target commutator value $ h\in [G, G] $, count and construct pairs $ (x, y) $ with $ xy = g $ and $ \left[x, y\right] = h $. In an explicit bilinear-data input model, we show that the decision, counting, and search variants reduce to solvability of a single linear system $ T_w(u) = h $ over $ \mathbb{F}_p $, where $ T_w $: $ u\mapsto b(u, w) $ is the contraction map determined by the $ V $-projection $ w $ of $ g $. When solvable, the solution set $ \mathcal{S}(g, h) $ is exhibited as an explicit torsor for $ \ker(T_w)\times W $, yielding a closed counting formula and a certified witness construction by elementary linear algebra. We derive exact secondary laws: a complete description of the attainable commutator set $ \mathcal{H}(g) = \operatorname{im}(T_w) $, an exact uniformity law over attainable values, and a factor-swap bijection relating $ \mathcal{S}(g, h) $ to a shifted product constraint. Finally, we define rank-profile polynomials $ P_G(t) = \sum_{w\in V} t^{\operatorname{rank}(T_w)} $, prove isomorphism invariance, and extract further invariants (radical size, extremal attainable-set size) directly from counting oracles. The odd-prime hypothesis is made explicit throughout: the centered coordinate law uses the scalar $ 1/2\in\mathbb F_p $, whereas characteristic two requires a cocycle or quadratic-refinement formulation. We also record coordinate-invariance, conversion costs from power-commutator input, sparse implementation refinements, and limitations of the rank-profile invariant.
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Ghaliah Alhamzi, Mdi Begum Jeelani, Wael Mahmoud Mohammad Salameh, Prakash Jadhav. Commutator-constrained factorizations in class-two exponent-p-groups[J]. AIMS Mathematics, 2026, 11(6): 15561-15580. doi: 10.3934/math.2026640
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