Computing in Direct Powers of Expanded Groups. A Discussion of the Subpower Intersection Problem

Contents

1 Universal Algebra and GAP

1.1 Algebras

1.2 GAP

1.3 An Example Combining Theory and GAP

2 Expanded Groups

2.1 Generators of Expanded Groups

2.2 Functions with Finite Degree

2.3 Direct Powers and Ideals of Expanded Groups

2.4 A Data Structure for Expanded Groups in GAP

2.5 Computing in Expanded Groups with GAP

3 Subpower Intersection Problem

3.1 Strong Generators of Groups

3.2 The Subpower Intersection Problem

3.3 Implementation in GAP

3.4 Counting Compatible Functions

Research Objectives and Core Topics

This master thesis explores the mathematical theory of expanded groups and develops algorithms to solve the Subpower Intersection Problem for direct powers of groups, with a focus on practical implementation within the GAP computer algebra system.

• Theoretical foundation of universal algebra and expanded groups.
• Development of robust data structures for expanded groups in GAP.
• Implementation and analysis of algorithms for strong generators.
• Solving the Subpower Intersection Problem using algorithmic approach.
• Practical application in counting compatible functions on groups.

Excerpt from the Thesis

Summary of Chapters

1 Universal Algebra and GAP: This chapter introduces fundamental concepts of universal algebra and provides an introduction to the GAP programming language.

2 Expanded Groups: This section details the theory of expanded groups, defines generators for them, and implements a dedicated data structure within GAP for computational purposes.

3 Subpower Intersection Problem: The final chapter addresses the Subpower Intersection Problem by introducing strong generators and providing a concrete implementation for solving these problems in GAP.

Keywords

Universal Algebra, Expanded Groups, Group Theory, GAP, Subpower Intersection Problem, Strong Generators, Polynomial Functions, Direct Products, Computational Algebra, Permutation Groups, Algorithm Implementation, Subpower Membership Problem, Compatible Functions, Normal Subgroups.

Frequently Asked Questions

What is the primary focus of this work?

The thesis focuses on the computational aspects of expanded groups and direct powers of groups, specifically aiming to solve the Subpower Intersection Problem using the GAP system.

What are the main thematic areas covered?

The work covers universal algebra, the theory of expanded groups, algorithmic implementation of strong generators, and applications like the Subpower Membership and Intersection Problems.

What is the core research objective?

The primary goal is to implement and analyze efficient algorithms for handling direct powers of expanded groups and their subpower properties within the GAP software environment.

Which scientific methodology is employed?

The methodology combines mathematical proof for algebraic theorems with algorithmic implementation and empirical verification through computational experiments in GAP.

What topics are discussed in the main part of the thesis?

The main part covers the theory of expanded groups, the construction of specific data structures for these objects, and the development of algorithms for strong generators and the intersection of subgroups.

How would you characterize this thesis with keywords?

Key terms include Universal Algebra, Expanded Groups, GAP, Subpower Intersection Problem, and Computational Algebra.

How is the Subpower Intersection Problem addressed?

The problem is addressed by constructing transversals (strong generators) using specialized sifting algorithms and closing these sets under product operations.

Why are expanded groups important in this research?

Expanded groups serve as the primary mathematical structure for studying more complex properties like polynomial equivalence and ideal structures beyond simple group operations.

What is the significance of the implemented GAP functions?

They enable the automated calculation of properties that are otherwise too complex to compute by hand, such as determining all binary polynomial functions for groups of order 8.