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Subgroups

This menu is the primary tool for exploring the structure of a group.

Under Find Subgroups, the form on the right displays. To find the subgroup generated by a collection of elements, check the boxes next to the elements, then click on Test for Closure.

The elements you must add for closure will flash in the table. Check the boxes next to the flashing elements, then Test for Closure again.

Repeat this process until there are no more flashing elements.



Create a New Table

 

When you add the elements necessary for a subgroup, its group table displays and it is added to your list of Found Subgroups, as illustrated on the right where 3 subgroups of D4 have been found.  

If you close one of the subgroup tables, you can display it again by selecting Show Found Subgroups in the form for Find Subgroups, or you can select an individual subgroup in the drop down menu on the upper right corner. 


Under Cyclic Subgroups, GU displays <a> sorted by powers of a. It also displays all cyclic subgroups.

Under Show All Subgroups, GU displays the subgroups arranged by order, with subgroups of the same order on the same line. 


Subgroups of D4
 

 

When you click on Subgroup Lattice, the lattice of the selected table displays. When you click on a subgroup on the lattice, its left cosets display, as illustrated on the right.

Your computer should be able to display the lattice for Z60 and A5, which have 60 elements. However, it may not be able to display the lattices for larger groups.  

 

 


 

When you click on Normal  Subgroups, a list of all subgroups displays. When you select one of the listed subgroups, its left and right cosets display, along with a message as to whether or not it is normal, as illustrated on the right.

 


Lattice for D4

        

 

There are two other ways to use GU to determine if a subgroup is normal.

Under Cosets, you can easily see whether or not the product of two cosets is a coset. When you select one of the listed subgroups, GU displays the original table sorted by its cosets and each coset has an identifying color.  The arrangement of the colors tells you whether or not the subgroup is normal. For example, in the adjacent table, the colored cells show that the product of two cosets is not always a coset, so the selected subgroup is not normal.

Under Factor Groups, all normal subgroups of the selected table display. When you select one of the listed subgroups, the table for its factor group displays along with a key for the cosets in the table, as illustrated in the adjacent table of a factor group of D6.

You can display the Center of a selected group and the Centralizer of a selected element.

Under Conjugates, you can display the conjugates of a selected element and a listing of the conjugacy classes.

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D6 Factor Group