# Sage Interactions - Groups Using the GAP System

This page was first created for Google Summer of Code 2021 by Alexis Newton, with mentorship by Mckenzie West.

If you have interactions that you are interested in adding to this page, please do so. You can also contact Alexis Newton at [email protected].

The main function of this page is to demonstrate the improved capabilities that we have developed for SageMath during the GSOC Summer 2021. The trac ticket for those developments can be found here: https://trac.sagemath.org/ticket/32196#comment:5.

Note that this ticket has not yet been approved, so in many cases we have defined our main function "order_n" in the Sage cell to be a more rudimentary version of the what is contained in the trac ticket. We do this in order to demonstrate how it may be used. Once the ticket has been pushed through the Sage development process, we will update the page to remove this extraneous code.

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Contents

## Calling Groups from GAP

GAP is a system for computational discrete algebra, which provides a programming language, a library of thousands of functions implementing algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects. GAP is used in research and teaching for studying groups and their representations, rings, rings, vector spaces, algebras, combinatorial structures, and more.

Using Sage, we are able to interface with the GAP System to call groups of different types. This is extremely useful for demonstrating examples in undergraduate abstract algebra courses. Learn more about interfacing with GAP via Sage here: https://doc.sagemath.org/html/en/reference/interfaces/sage/interfaces/gap.html.

We rely heavily on the Small Group and All Small Groups GAP commands within this page. The documentation for these can be found on the GAP System website: https://www.gap-system.org/Manuals/pkg/SmallGrp/doc/chap1.html.

### Group of Order n

Use this interact to call a group of order n from the GAP library.

### Group of Order n of a Certain Type

Use this interact to specify a type of group to call.

### All Groups of Order n

Use this interact to call all groups of order n from the GAP library.

### All Groups of Order n of a Certain Type

Use this interact to specify a type of group to call.

### Small Group Info

Use this interact to learn information about the small groups of order n contained in the GAP library.

## Calling a List of Groups from GAP

The current configuration for calling information from GAP only allows for you to call one group at a time, but we aim to allow for lists of groups which meet certain parameters. The some of the group properties we aim to address include abelian, solvable, nilpotent, given order, dihedral, semi-direct products, alternating, symmetric, and simple.

### Groups Order Less Than or Equal to n

Use this interact to call all groups from the GAP library that have order less than or equal to your desired value.

### Groups Order Between m and n

Use this interact to call all the groups from the GAP library that have order between m and n

### Groups of a Certain Type

Use this interact to call all groups of a certain type from the GAP library that have order between m and n

### Direct or Semidirect Product Groups

Use this interact to specify groups that contain direct or semidirect products.

* WARNING:* This code uses the StructureDescription function, but this is not an invariant on groups through gap: non-isomorphic groups can give different structure descriptions. Also, a group may have an x in the structure description but not actually be a direct product (via John Sawatzky).