Word Length and Automorphism-Based Element Classification in the Discrete and Finite Heisenberg Group The Exponentiated Power Shanker Distribution With Application
Rodman F. Manalang and Melvin A. Vidar (15-35)
Abstract
The Heisenberg group over Z, HH(Z), is the set of all 3 × 3 upper triangular matrices with integer entries above the diagonal and ones on the diagonal together with matrix multiplication. For a positive integer nn ≥ 2, its finite counterpart, HH!, is defined similarly with entries in Z! under modular multiplication. Equivalently, these groups can be described as sets of triples (aa, bb; cc) with a nonabelian group operation involving the standard commutator structure.
This paper investigates the word length, the minimal number of generators required to express a group element in HH(Z) and HH”, where pp is prime, relative to their standard generating sets. We define and analyze two specific automorphisms, σσ and φφ, that preserve word length and enable us to limit our analysis to a smaller representative subset of group elements without loss of generality. These automorphisms allow for more efficient classification and the derivation of explicit formulas for computing word length. We use these results to categorize elements in HH” into three types based on their algebraic structure and their behavior under these automorphisms. In addition, we explore identities related to word length.
The results reveal new insights into the combinatorial and algebraic structure of Heisenberg groups, with implications for computational group theory, representation theory, and cryptography. Our approach offers a novel perspective that bridges structural group theory with algorithmic word computation.