group visualizer

Group theory is a wonderful branch of mathematics! It has many interesting results, and offers a variety of graphical representations.

This page was developed alongside our video series about groups. It offers a place to explore small groups, through three methods of visualization:

particles

In this visualization, each element is represented by a particle. An 'operating element' can be chosen to link these particles with arrows. For example: if the group is addition of integers and 3 is chosen as the operating element, then there will be an arrow from 0 to 3, from 1 to 4, from 2 to 5, and from every n to (n+3).

For a more detailed graph, you can select multiple operating elements, denoted in different colors. Each operating element has a 'strength' which is set using the slider. A high strength causes linked particles to have strong attraction, while a low strength causes weak attraction. Vary these strengths to see the effect on the graph.

cayley diagrams

The particle graphs are technically Cayley Diagrams, but they can be messy. For a more structured view, the elements are fixed in a regular pattern. This dispalys a more typical Cayley Diagram - a visual of all elements linked by arrows determined by the operating elements. The cycles rotate to offer changing perspectives on the graph, but no attraction between elements is applied.

tables

The most comprehensive visualizations are tables. These show the result from operating any pair of elements.

navigation

Use the icons to switch between visualizations:

Use the arrows below these icons to select the group. There are 3 levels of granularity. First, set the order to determine the number of elements in the group. Second, set the class to determine the group, up to isomorphism. Third, set the group to determine the representation of that particular group.

Use the other control box to set operating elements. Click the colored circles to enable or disable them. The white ring denotes which operating element you are currently editing.

about me

I am TheGrayCuber, a creator of mathematical websites and videos!

My fascination with group theory began with the multiplicative groups mod n. It is so cool that finite multiplication is just finite addition. I created a page that explored the relationships between these groups, using a particle system similar to this page, though each particle represented an entire group rather than just a single element.

I was further inspired by a section of Contemporary Abstract Algebra by Joseph A. Gallian, which classified all groups with order 1-15. I love classification problems, and tried to continue with 16, but immediately learned that the book ended with 15 for good reason. Order 16 has a lot of groups, and they're much trickier than 1-15.

Working on this page and our video series helped me to fully understand those groups of order 16 and to continue the classification problem to higher orders! It was also a much needed push to figure out semi-direct products.

The first draft of this page was quite experimental. I made the controls modular - everything could be resized, recolored, and dragged around to any position. Eventually I decided that it wasn't worth the hassle. This would mainly just help me to format the page for a video. It wouldn't matter to the average user. But it wasn't all in vain - that system for designing and positioning controls became the starting point for TheGraySlides, the program that I now use to make all of my videos!


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