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

Date of Publication


Document Type


Degree Name

Doctor of Philosophy in Mathematics

Subject Categories



College of Science


Mathematics and Statistics Department

Thesis Adviser

Melvin A. Vidar

Defense Panel Chair

Arlene A. Pascasio

Defense Panel Member

Reginaldo M. Marcelo
Ederlina G. Nocon
Edmundo D. Perez, Jr.
Diana C. Songsong


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.

Abstract Format






Accession Number


Shelf Location

Archives, The Learning Commons, 12F Henry Sy Sr. Hall

Physical Description

1 computer disc; 4 3/4 in.


Ordered algebraic structures; Algebra

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