PascGalois JE Help: Group Calculator


The group calculator is a simple facility to help the user do group operations. While you can probably do calculations in Zn faster in your head than type them into this calculator you may find it handy when working with a dihedral or permutation group.

The calculator is broken down into three parts. The top is for operating on two group elements and for taking powers of a single element. There is also a feature for interchanging the two entries for a quick check of commutativity. The middle contains a facility for generating a subgroup. The bottom section allows the user to generate a coset of the subgroup generated in the middle.

The Group Operation Section

To do a group operation on two group elements simply type the two elements into the text boxes and click on the "*" tool. The result of the operation will be displayed in the disabled text box after the equal sign. To take an element to a power type the element in the first box and the power in the second and select the "^" tool. In this case the second text box must contain an integer. In the event that an element does not have an inverse, which could happen if a user-defined structure is involved or if you are using Zn under multiplication, an error message will be displayed. The final tool simply interchanges the two text boxes.

The Subgroup Section

To generate a subgroup simply type in the elements you wish to use as generators with a comma between each and then click the generate subgroup tool. You may use < > around the element list but it is not necessary.

The toolbar consists of six options.

The Cosets Section

To generate a coset of the above subgroup simply type in a coset representative and click either the left coset button or the right coset button. If the subgroup has not been generated the coset generator will generate the subgroup first and then the coset. Note that when generating a subgroup the elements of the subgroup will be sorted. As for the coset, we do not sort the elements but leave them in the order calculated. This way the user can see the correspondence between the subgroup list and the coset list.

The element list toolbar has six options.