Groups Unlimited   -  a new dimension for learning Group Theory   



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GU provides an easy-to-use, colorful interface where students can create, compare, and explore groups in a variety of ways that will deepen their understanding and appreciation of the basic concepts of group theory.

GU also provides an easy way for teachers to demonstrate basic properties of groups in class.

Best of all, no special instructions are needed to learn how to use the program. GU is designed to be very user-friendly and visually appealing.


With GU, students can easily create multiple groups via entries in a blank table or open a group from the group library, which contains menus for the following :

  • Zn - additive mod n groups
  • Un - multiplicative mod n groups
  • Dn - dihedral groups
  • Sn - permutation groups
  • An - alternating groups
  • Product Groups
  • Library of groups up to order 40

Dihedral group tables display with a regular polygon that students can flip and rotate to illustrate the operations.

                      To see D5, click here.         


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Students can explore, compare, and contrast the structure of different groups through easy-to-use menus for finding cyclic subgroups and subgroups generated by a subset, or listing all subgroups, whose internal connections can be beautifully illustrated with a colorful lattice.  

To see the lattice for S4, click here.

By comparing lattices, students can see whether or not two groups of the same order might be isomorphic. They can then use the homomorphism menu to define functions between two groups and determine whether or not they preserve the operation.



With GU, students can:

  • Determine if a subgroup is normal by listing all left and right cosets, or arranging the table by colored cosets, as illustrated on the right.

  • Compute tables for factor groups and compare them with other groups in the library.

  • Analyze permutation tables in condensed letter form or display entries in cycle notation.

  • Compare product tables in condensed letter form or display entries as ordered pairs.

GU also has explorations where students can discover and illustrate properties of groups.


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