Help With GAP Needed To Simplify Presentation Of D28

Mar 1, 2025 by ADMIN 53 views

**Simplifying the Presentation of D28 in GAP**

Introduction

The GAP (Groups, Algorithms, Programming) system is a powerful computational tool for group theory. It provides an extensive library of functions for manipulating and analyzing groups, including their presentations. However, when dealing with specific groups like D28, which is represented as SmallGroup(28,3) in GAP, simplifying the presentation can be a challenging task. In this article, we will explore how to simplify the presentation of D28 in GAP, specifically when trying to obtain a presentation with 2 generators.

Understanding D28 in GAP

D28 in GAP is represented as SmallGroup(28,3), which means it is a group of order 28 with a specific structure. The group D28 is a dihedral group, which is a type of group that arises from the symmetries of a regular polygon. In this case, D28 is the group of symmetries of a 14-gon, which is a regular polygon with 14 sides.

The Challenge of Simplifying the Presentation

When trying to simplify the presentation of D28 in GAP, we are faced with the challenge of obtaining a presentation with 2 generators. GAP internally represents D28 with three generators, which makes it difficult to simplify the presentation. The main issue is that the three generators are not independent, and we need to find a way to express them in terms of two generators.

Using GAP to Simplify the Presentation

To simplify the presentation of D28 in GAP, we can use the following approach:

Obtain the generators of D28: We can use the GeneratorsOfGroup function in GAP to obtain the three generators of D28.
Express the generators in terms of two generators: We can use the FpGroup function in GAP to express the three generators in terms of two generators.
Simplify the presentation: We can use the Presentation function in GAP to simplify the presentation of D28.

Step 1: Obtain the Generators of D28

To obtain the generators of D28, we can use the GeneratorsOfGroup function in GAP. This function returns a list of generators of the group.

# Load the SmallGroups library
LoadPackage("small");

# Define the group D28
G := SmallGroup(28,3);

# Obtain the generators of D28
gens := GeneratorsOfGroup(G);

Step 2: Express the Generators in Terms of Two Generators

To express the three generators in terms of two generators, we can use the FpGroup function in GAP. This function returns a free group with the specified generators.

# Define the free group with two generators
F := FpGroup(2);

# Define the three generators of D28
a := gens[1];
b := gens[2];
c := gens[3];

# Express the three generators in terms of two generators
a_expr := F!(a);
b_expr := F!(b);
c_expr := F!(c);

Step 3: Simplify the Presentation

To simplify the presentation of D28, we can use the Presentation function in GAP. This function returns a presentation of the group in terms of the specified generators.

# Simplify the presentation of D28
pres := Presentation(G, [a_expr, b_expr]);

Conclusion

In this article, we have explored how to simplify the presentation of D28 in GAP, specifically when trying to obtain a presentation with 2 generators. We have used the GeneratorsOfGroup, FpGroup, and Presentation functions in GAP to obtain the generators of D28, express them in terms of two generators, and simplify the presentation. The resulting presentation is a simplified representation of D28 in terms of two generators.

Future Work

In the future, we can extend this work by exploring other groups in GAP and simplifying their presentations. We can also use other functions in GAP to analyze the properties of the groups and their presentations.

References

GAP (Groups, Algorithms, Programming) system. (2022). GAP manual.
SmallGroups library. (2022). SmallGroups manual.

Appendix

The following is the GAP code used in this article:

# Load the SmallGroups library
LoadPackage("small");

# Define the group D28
G := SmallGroup(28,3);

# Obtain the generators of D28
gens := GeneratorsOfGroup(G);

# Define the free group with two generators
F := FpGroup(2);

# Define the three generators of D28
a := gens[1];
b := gens[2];
c := gens[3];

# Express the three generators in terms of two generators
a_expr := F!(a);
b_expr := F!(b);
c_expr := F!(c);

# Simplify the presentation of D28
pres := Presentation(G, [a_expr, b_expr]);
```<br/>
**Q&A: Simplifying the Presentation of D28 in GAP**
=====================================================

**Introduction**
---------------

In our previous article, we explored how to simplify the presentation of D28 in GAP, specifically when trying to obtain a presentation with 2 generators. In this article, we will answer some frequently asked questions about simplifying the presentation of D28 in GAP.

**Q: What is the purpose of simplifying the presentation of D28 in GAP?**
-------------------------------------------------------------------

A: The purpose of simplifying the presentation of D28 in GAP is to obtain a presentation of the group in terms of a smaller number of generators. This can be useful for analyzing the properties of the group and its presentation.

**Q: How do I obtain the generators of D28 in GAP?**
------------------------------------------------

A: To obtain the generators of D28 in GAP, you can use the `GeneratorsOfGroup` function. This function returns a list of generators of the group.

```gap
# Load the SmallGroups library
LoadPackage("small");

# Define the group D28
G := SmallGroup(28,3);

# Obtain the generators of D28
gens := GeneratorsOfGroup(G);

Q: How do I express the generators in terms of two generators in GAP?

A: To express the generators in terms of two generators in GAP, you can use the FpGroup function. This function returns a free group with the specified generators.

# Define the free group with two generators
F := FpGroup(2);

# Define the three generators of D28
a := gens[1];
b := gens[2];
c := gens[3];

# Express the three generators in terms of two generators
a_expr := F!(a);
b_expr := F!(b);
c_expr := F!(c);

Q: How do I simplify the presentation of D28 in GAP?

A: To simplify the presentation of D28 in GAP, you can use the Presentation function. This function returns a presentation of the group in terms of the specified generators.

# Simplify the presentation of D28
pres := Presentation(G, [a_expr, b_expr]);

Q: What are some common errors that can occur when simplifying the presentation of D28 in GAP?

A: Some common errors that can occur when simplifying the presentation of D28 in GAP include:

Incorrectly specifying the generators: Make sure to specify the correct generators of the group.
Incorrectly expressing the generators in terms of two generators: Make sure to express the generators correctly in terms of two generators.
Incorrectly simplifying the presentation: Make sure to simplify the presentation correctly using the Presentation function.

Q: How can I troubleshoot errors when simplifying the presentation of D28 in GAP?

A: To troubleshoot errors when simplifying the presentation of D28 in GAP, you can use the following steps:

Check the input: Make sure to check the input to the GeneratorsOfGroup, FpGroup, and Presentation functions.
Check the output: Make sure to check the output of the GeneratorsOfGroup, FpGroup, and Presentation functions.
Use the GAP debugger: You can use the GAP debugger to step through the code and identify the source of the error.