Some combinatorial and algebraic structures in the discrete and finite Heisenberg groups

• Main Author: Manalang, Rodman Flores
• Format: text
• Language: English
• Published: Animo Repository 2018
• Subjects: Ordered algebraic structures; Algebra; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etd_doctoral/555
• Institution: De La Salle University

The discrete Heisenberg group, H(Z), is the set of all 3×3 upper triangular matrices whose diagonal entries are all 1 and whose entries above the diagonal are integers under matrix multiplication. Whereas for a positive integer n ≥ 2, the finite Heisenberg group, Hn, is the set of all 3×3 upper triangular matrices with 1′s in the diagonal and with entries above the diagonal coming from Zn under matrix multiplication mod n. It is known that H(Z) and Hn have a standard generating set S =   X =   1 1 0 0 1 0 0 0 1   , Y =  1 0 0 0 1 1 0 0 1     . Thus, for any element g ∈ H(Z) (respectively Hn), there exists a nonnegative integer k such that g = m±1 1 m±1 2 ...m±1 k , mi ∈ S, (1 ≤ i ≤ k). The wordlength of an element g with respect to the standard generators is the minimum value of k that satisfies the above equation. In this dissertation, we present a construction of automorphisms, σ and φ, of H(Z) and Hp (p is prime) that preserves wordlength. We will show how these mappings can be used to reduce the domain with respect to its wordlength. Main results include the combinatorial formulas for the wordlength of the elements of the discrete and finite Heisenberg groups. Furthermore, we use σ to classify the elements of Hp and obtain some algebraic structures.