@article{Alhamzi2026, 
author = {Ghaliah Alhamzi and Mdi Begum Jeelani and Wael Mahmoud Mohammad Salameh and Prakash Jadhav},
title = {Commutator-constrained factorizations in class-two exponent-p-groups},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {15561-15580},
keywords = {computational group theory, nilpotent p-groups, class-two groups, Baer correspondence, commutator map, exact counting, linear algebra over finite fields, isomorphism invariants},
url = {https://www.sciopen.com/article/10.3934/math.2026640},
doi = {10.3934/math.2026640},
abstract = {Groups of nilpotency class    2 and exponent    p (with    p odd) admit concrete coordinate models governed by alternating              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  ∈  G and a target commutator value    h  ∈  [  G  ,  G  ], count and construct pairs    (  x  ,  y  ) with    x  y  =  g and        [          x      ,      y        ]    =  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              F        p  , where        T    w  :    u  ↦  b  (  u  ,  w  ) is the contraction map determined by the    V-projection    w of    g. When solvable, the solution set        S    (  g  ,  h  ) is exhibited as an explicit torsor for    ker  ⁡  (      T    w    )  ×  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        H    (  g  )  =  im  ⁡  (      T    w    ), an exact uniformity law over attainable values, and a factor-swap bijection relating        S    (  g  ,  h  ) to a shifted product constraint. Finally, we define rank-profile polynomials        P    G    (  t  )  =      ∑          w      ∈      V            t          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  ∈            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.}
}