Group Explorer 3.0 Help: Subset Options

Every visualizer in Group Explorer except objects of symmetry have a panel like the one shown below for examining subgroups of the group under study.

The particular subsets panel shown above is for the eight-element quaternion group.

Tour of the subsets pane

The topmost portion of the interface lists all the subgroups of the group shown in the main portion of the view, beginning with $H_0$, the trivial (one-element) subgroup, and ending with $H_n$, the whole group, for a group with $n+1$ subgroups. Group Explorer computes all subgroups of each group when loading it; the user cannot add or remove any entries from this section.

The next portion of the window lists user-defined subsets, which is empty by default. You can add and delete subsets relevant to your study of the group; see below.

The last portion of the window lists partitions of the group (e.g. conjugacy classes, cosets of a particular subgroup, etc.) and is particularly useful for creating informative highlightings of the visualizer. Partitions can be added to this section via various computations; see below.

The subsets panel allows computations and highlighting with respect to both subsets and subgroups of whatever group is being visualized. It has many features, covered one at a time below.

For any subset listed in the pane, you can find out its list of elements by double-clicking [touch and holding] the subset you’re interested in.

Right-clicking [tapping] on the subsets pane brings up a menu from which you can take several important actions, some of which are visible in the following screenshot. You can pin a submenu to keep it from disappearing by clicking [tapping] on its parent menu item (the line in the parent menu with a right-pointing triangle, like “Compute ►” in the following screenshot); you can clear it by clicking [tapping] on the line again.

Creating and deleting user-defined subgroups are both possible using this menu, as well as several types of computation and highlighing. Here follows the documentation for each option on this menu.

Topmost items on the popup menu

Edit list of elements in $S_i$

This menu item appears only if you selected a user-defined subset. If you choose it, a window like the one pictured here opens.

The left pane shows the elements in your subset and the right pane shows the other elements of the group. To add elements to your subset, drag them in from right to left. Dragging from the left pane to the right pane takes elements out of your subset. OK approves your changes and Cancel discards them.

Delete $S_i$

This menu item appears only if you selected a user-defined subset. Choosing it deletes the user-defined subset you selected.

Delete $CC_i$, $OC_i$, $gH_i$, etc.

This menu item appears only if you selected a partition (i.e. on one of its sets). Partitions can be conjugacy classes $CC_i$, order classes $OC_i$, left cosets $gH_i$, or right cosets $H_ig$. Choosing this option deletes the entire partition you selected (e.g. all conjugacy classes, all order classes, or all left/right cosets of the subgroup, etc.).

Create $S_i$

This menu item creates a new subset under the “User-defined subsets” heading. The subset will be empty, and you can add elements to it as described above.

The Compute > submenu

Compute > all conjugacy classes $CC_i$

This submenu item appears only if the conjugacy class partition does not already exist. It adds the set of conjugacy classes as a new partition under the “Partitions” heading.

Compute > all order classes $OC_i$

This submenu item appears only if the order class partition does not already exist. It adds the set of order classes as a new partition under the “Partitions” heading.

Compute > normalizer of $H_i$, $\text{Norm}(H_i)$

This submenu item appears when you select the subgroup $H_i$. It adds the normalizer of the given subgroup to the list of user-defined subsets.

Note that whenever you add a new subset, Group Explorer always checks whether it exists under another name, and gives you the option to cancel your addition if so.

Compute > closure of $X$, $\langle X \rangle$

This submenu item appears when you select the subset $X$, but only if the subset you selected is not a subgroup. It adds the closure of the given subset to the list of user-defined subsets.

Note that whenever you add a new subset, Group Explorer always checks whether it exists under another name, and gives you the option to cancel your addition if so.

Compute > left cosets $gH_i$

This submenu item appears when you select the subgroup $H_i$, but only if the partition by left cosets of that subgroup does not already exist. It adds the set of left cosets as a new partition under the “Partitions” heading.

Compute > right cosets $H_ig$

This submenu item appears when you select the subgroup $H_i$, but only if the partition by right cosets of that subgroup does not already exist. It adds the set of right cosets as a new partition under the “Partitions” heading.

Compute > an intersection > subsubmenu

This submenu appears when you select the subset $X$, and it contains items allowing you to perform an intersection of $X$ with any other subset listed in the whole pane. Choosing one of the items on this submenu computes the intersection described in that item, e.g. “the intersection of $H_2$ with $S_5$.” Intersection here means simply what it does in set theory–the elements in common between the two sets.

Note that whenever you add a new subset, Group Explorer always checks whether it exists under another name, and gives you the option to cancel your addition if so.

Compute > a union > subsubmenu

This submenu appears when you select the subset $X$, and it contains items allowing you to perform a union of $X$ with any other subset listed in the whole pane. Choosing one of the items on this submenu computes the union described in that item, e.g. “the union of $H_2$ with $S_5$.” Union here means simply what it does in set theory–the combined elements from the two sets.

Note that whenever you add a new subset, Group Explorer always checks whether it exists under another name, and gives you the option to cancel your addition if so.

Compute > an elementwise product > subsubmenu

This submenu appears when you select the subset $X$, and it contains items allowing you to perform an elementwise product of $X$ with any other subset listed in the whole pane. Choosing one of the items on this submenu computes the elementwise product described in that item, e.g. “the elementwise product of $H_2$ with $S_5$.”

Note that whenever you add a new subset, Group Explorer always checks whether it exists under another name, and gives you the option to cancel your addition if so.

Highlight subset by > submenu

This submenu appears only if you select a subset (as opposed to a heading). Its items depend on the visualizer as follows.

• For Cayley diagrams, highlighting options are
  Node color,
  Ring around node, and
  Square around node.

• For multiplication tables, highlighting options are
  Background,
  Border, and
  Corner.

• For cycle graphs, highlighting options are
  Background,
  Border, and
  Top.

Choosing one causes the subset you selected to be highlighted in the visualizer (both its large and small incarnations) with the method you selected. For instance, here is a multiplication table with an order-4 subgroup highlighted according to corners.

Note that this undoes any other highlighting of the type in question (e.g. former “Node color” highlighting will evaporate when new “Node color” highlighting is chosen).

Note further that in multiplication tables, highlighting items by background highlighting removes the default coloration scheme (e.g. rainbow, grayscale).

You can undo this highlighting using the “Clear all highlighting” option on this same menu (see below).

Highlight partition by > submenu

This submenu appears only if you select a partition (i.e. on one of its sets). Its items depend on the visualizer in the same way described above.

Choosing one of its items causes the partition you selected to be highlighted in the visualizer (both its large and small incarnations) with the method you selected, using a different color for each set in the partition. For instance, here is a cycle graph with the conjugacy class partition highlighted according to background.

Note that this undoes any other highlighting of the type in question (e.g. former “Node color” highlighting will evaporate when new “Node color” highlighting is chosen).

Note further that in multiplication tables, highlighting items by background highlighting removes the default coloration scheme (e.g. rainbow, grayscale).

You can undo this highlighting using the “Clear all highlighting” option documented immediately below.

Clear all highlighting

Removes all highlighting of any type from the visualizer. (Types are listed above.)