This page is a dictionary of terms specific to Group Explorer. Many other pages in the Group Explorer help link here to define terms. Unlike the group theory terminology page, these terms are not well-known mathematical terms; they’re used only in Group Explorer.
The files which store group information are called group files, and they can contain information about the original author of the file. This person encoded the description of a finite group into Group Explorer’s group definition syntax so that Group Explorer could load and manipulate the group. See also URL of a group.
The date of last modification of a group actually refers to the date on which the file from which the group was loaded was last modified. See also URL of a group.
Groups are stored on the Group Explorer website in files that end in the
extension .group. The group’s URL is the web address (beginning with
http://) pointing to the .group file from which the group was loaded on
Group Explorer’s website. Example:
https://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/Z_2 x Z_4.group
See representation of a group.
In order to display the elements of a group on the screen, Group Explorer needs to know their names. Although internally, Group Explorer stores groups in a manner consistent with the mathematical abstractions that they are, users prefer a prettier format. Each group file defines at least one representation, or naming scheme–that is, a list of names, one for each element of the group. Users can add additional representations (also called naming schemes) by using the controls in the group info page.
Note that group elements’ representations should not be confused with group presentations, which are embeddings of arbitrary groups into groups of matrices. Group Explorer does not currently have any features related to group presentations.
Sheets are a blank canvas on which the user can drop illustrations of a group, homomorphisms to connect them, and pieces of text for description. Thus groups need not be examined only in isolation; they can be compared to other groups.
To open a new sheet, from the main page, click the sheet icon on the top right. For more information, see the introduction to sheets or the reference documentation on the sheet interface.
Group Explorer uses the term visualizer to describe any of the various mechanisms for obtaining pictures of a group. For instance, one way to visualize a group is through its multiplication table, so we refer to multiplication tables as “visualizers.” Group Explorer contains four types of visualizers: multiplication tables, cycle graphs, cayley diagrams, and objects of symmetry.