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

## Date of Publication

2018

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy in Mathematics

## Subject Categories

Mathematics

## College

College of Science

## Department/Unit

Mathematics and Statistics

## 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

## Abstract/Summary

The discrete Heisenberg group, H(Z), is the set of all 3Ã—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 â‰¥ 2, the finite Heisenberg group, Hn, is the set of all 3Ã—3 upper triangular matrices with 1â€²s 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 = ï£±ï£´ï£² ï£´ï£³ X = ï£« ï£¬ï£ 1 1 0 0 1 0 0 0 1 ï£¶ ï£·ï£¸ , Y = ï£« ï£¬ï£1 0 0 0 1 1 0 0 1 ï£¶ ï£·ï£¸ ï£¼ï£´ï£½ ï£´ï£¾ .

Thus, for any element g âˆˆ H(Z) (respectively Hn), there exists a nonnegative integer k such that g = mÂ±1 1 mÂ±1 2 ...mÂ±1 k , mi âˆˆ S, (1 â‰¤ i â‰¤ 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, Ïƒ and Ï†, 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 Ïƒ to classify the elements of Hp and obtain some algebraic structures.

## Abstract Format

html

## Language

English

## Format

Electronic

## Accession Number

CDTG007630

## Shelf Location

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

## Physical Description

1 computer disc; 4 3/4 in.

## Keywords

Ordered algebraic structures; Algebra

## Recommended Citation

Manalang, R. F. (2018). Some combinatorial and algebraic structures in the discrete and finite Heisenberg groups. Retrieved from https://animorepository.dlsu.edu.ph/etd_doctoral/555