Group Explorer 3.0 Help: Editing Homomorphisms

If you double-click on a homomorphism in a sheet, you will be presented with the homomorphism edit dialog, like the one shown below.

Display of a homomorphism

The top of the homomorphism edit window allows you to customize the appearance of the homomorphism as an object in a sheet.

Morphism name:

This is filled with some default text, which will be shown in a small rectangular label on the morphism arrow in the sheet, as shown below.

The morphism name is displayed as HTML, so, as described here, if you would like the morphism name to appear as “f₂“, you may enter “f₂”, and the following will be displayed:

Show domain and codomain

If the checkbox beneath it, “Show domain and codomain,” is checked, then the label will be slightly larger, including the specified information, as shown below.

Show defining pairs

This box is not checked by default, but when checked, it augments the label with a list of all generating pairs, as in the “Mapping” portion of the dialog, discussed below. See the example below.

Show injective/surjective

This box is not checked by default, but when checked, it augments the label as follows:

• If the homomorphism is injective, then the phrase “1-1” is added (because “one-to-one” is a shorthand for “injective”). Otherwise, the phrase “not 1-1” is added.
• If the homomorphism is surjective, then the phrase “onto” is added (because “onto” is a shorthand for “surjective”). Otherwise, the phrase “not onto” is added.

Two of these phrases appear in the example shown below.

Draw multiple arrows

This box is not checked by default, but when checked, it replaces the single arrow of the morphism with several arrows, one for each element of the domain. The source of these arrows is dependent on the visualizer; for instance, in the multiplication table below, the left column of the multiplication table is used. The endpoint of the arrows is determined by the map; in the example below, the map $z$ is the zero map, mapping all elements to the identity.

Arrows margin

The arrows margin adjustment determines how far from the center of the visualizers’ representations of the source and target group elements the arrows should begin and end. This can keep the arrowheads from obscuring the target element name in a complicated group. The units represent the percentage of the distance between the two elements to move the arrowheads away from the target. The number and the slider are synchronized, as described here.

Mathematical definition of the homomorphism

The bottom half of the homomorphism edit window allows you to specify the homomorphism as a function, a map from one group to another.

CITE(VGT-8 MM-4 DE-9 TJ-9 TJ-11)

Pairs in the map

Under the heading “Define the homomorphism here,” we have a table listing which elements of the homomorphism’s domain map to which elements of its codomain. Each row of the table is a pair of elements, call them $d$ from the domain and $c$ from the codomain, that have been added to the homomorphism already, meaning that the user has stipulated that $f(d)=c$ (if the morphism is named $f$).

Next to each pair in the map appears a “Remove” button so that if you wish to edit the morphism by mapping $d$ to a different element, you can remove the pair $(d,c)$ from the map and then assign $d$ a new image, as documented below.

The user assigns a pairing $(d,c)$ by selecting $d$ from the domain drop-down list beneath the table and $c$ from the codomain list, then clicking the “Add:” button. Because of the way the drop-down lists are arranged, they will read like the equation $f(d)=c$ before you click “Add,” to ensure that you’re adding the pair you intend.

The domain drop-down list contains only elements whose image has not already been determined by the pairs that have been added to the map. For instance, if you have already placed $(a_1,b_1)$ and $(a_2,b_2)$ into the map, then by the properties of homomorphisms, the image of all elements of the subgroup $\langle a_1,a_2 \rangle$ have been determined. Thus none of those elements will show up in the domain drop-down list, because none are assignable any longer.

When the domain elements of the pairs in the map don’t generate the entire domain, Group Explorer searches for any homomorphism extending the pairs in the table and uses that. The display window is continuously updated with the current mapping, including Group Explorer’s choices for unmapped elements, so the arrows shown always represent the current definition. (You may need to select “Draw multiple arrows” to see the full details.) You can also click on “Full morphism mapping” to see a synchronized display of the entire homomorphism in tabular form. Then you can simply continue to add pairs to the map to specify exactly the one you want.

The codomain drop-down list contains only elements that could legitimately be the image of the element chosen in the domain drop-down list. By “could legitimately be the image,” I mean that there exists some homomorphism that contains the pairs already added to the map plus the new pair you might add. Thus this constraint ensures that you can use the editor to define only homomorphisms, not any other kind of function between groups.

Other buttons

The “OK” button saves your changes to the morphism and closes the dialog.

The “Cancel” button closes the dialog without saving your changes into the sheet.

The “Delete morphism” button deletes the morphism and closes the dialog.