Editing Cayley Diagrams
Note: This page has a twin at Editing
Multiplication Tables.
Outline
The purpose of this page is twofold. First, we give a detailed
account of the uses of the Edit Cayley Diagram dialog box. The Edit
Cayley Diagram dialog box is a window with five tabbed sheets, each of
which has several controls on it. For this reason, it has
significant complexity, which can be dizzying. To clarify the use of
each control in the window, we handle each tabbed page at a time.
Second, we give an example of using these tools to exhibit the normality
of a subgroup within the currently loaded group. Here follows our
outline.
 | Tab 1: "Define H and a" - For selecting a subgroup H
and an element a within the group |
| Tab 2: "Highlight" - For using shape or color to visually
indicate properties of some elements in the group |
| Tab 3: "Generators" - For changing the way the diagram is
generated from the group elements |
| Tab 4: "Axes & Priority" - For changing the way the diagram
is laid out in space, and which generators take precedence |
| Tab 5: "Chunking & Arrows" - For choosing which arrows should
appear in the diagram, and for drawing translucent chunking boxes around
cosets of H |
| Example - Exhibiting normal subgroups visually by editing
Cayley diagrams |
The tabs are arranged from left to right in the window in an order that
the user could sensibly follow in customizing options. Although the
user is free to use any one tab they choose at any time, the order of the
tabs can help delineate the dependencies of some options on earlier
choices. Tabs more to the right tend to depend on choices made in
earlier tabs--those more to the left.
Each tab in the window has at its top an instructions panel, which
contains a (sort of) succinct set of directions for that tabbed page. We quote
each of those sets of directions below, and expound upon them.
Here is one view of the first tab in the Edit Cayley Diagram dialog box.
Its instructions panel contains the following text (only part of which is
visible in the picture above). We add to these directions below.
Several aspects of Cayley Diagram generation and
highlighting can be customized based on user-selected elements or
subgroups.
This page allows you to define a subgroup H and an element a within
the current group. The Highlight page and the Axes & Priority
page both contain controls that refer back to the subgroup H and the
element a selected on this page.
To define a subgroup H, you can use one of two methods:
1. Create H manually.
To manually add generators to and remove generators from the subgroup
H, follow these directions:
a. In the list next to "Define the subgroup H to be:" choose the
option "the subgroup defined manually below."
b. Choose an element from the list next to the "Add this element to H"
button.
c. Click the "Add this element to H" button to add the selected
element to the list of generators for H. It will appear in the textual
definition of H, between the <angle brackets>.
d. Repeat steps b. and c. to add additional generators to H, if
desired.
e. To remove a generator from H, select it from the list next to the
"Remove this element from H" button, and click the button.
2. Choose a predefined H.
To choose a subgroup H that is calculated for you by Group Explorer,
do the following:
a. In the list next to "Define the subgroup H to be:" choose an option
other than "the subgroup defined manually below."
b. The controls related to adding generators to and removing
generators from H will become gray, indicating that they are not
active. The textual definition of H will contain a list of generators
for the subgroup you selected.
To define an element a from the group, you simply select it from the
drop-down list at the bottom of the page.
Note: Clicking "Last highlighting" in the "Highlight" page can alter
the values of H and a on this page. See the instructions on that page
for details. |
Defining a
Thus the bulk of the first tabbed page is dedicated to defining a subgroup
H, and only the bottom line (reading "Define the element a to be:")
is given to specifying the element a. The drop-down list for
defining a contains each element in the group, shown according to
the representation chosen on the
Group menu.
Defining H
The first control that appears below the instructions pane is labeled "Define the subgroup
H to be," and contains three options.
| the subgroup defined manually below - Choosing this option
indicates that Group Explorer should attend to the rest of the controls
for defining H manually, as opposed to making it one of the following two
automatically computed groups |
| the commutator subgroup - Choosing this option indicates that
Group Explorer should ignore the rest of the controls for defining H manually,
and rather let H be the commutator subgroup of the currently
loaded group. That is, let H = { aba-1b-1
| a,b in H }. |
| the group's center - Choosing this option indicates that
Group Explorer should ignore the rest of the controls for defining H manually,
and rather let H be the center of the currently loaded group.
That is, let H = { a in H | for all b in
H, ab=ba }. |
Then follow a set of controls for manipulating the elements of H,
provided that the "defined manually" option was chosen above.
Adding elements to H: In the drop-down list next to
the "Add this element to H" button, one will find a list of all elements
of the group that are not already in the subgroup H. Because
H starts out empty, this drop-down list starts with every group
element in it. As above, element names are written according to the
representation chosen on the
Group menu. To add an element to H, choose it from the
drop-down list, and click the "Add this element to H" button. When
you do so, two changes will occur. First, the definition of H,
which starts out reading "H = < >" to indicate that H is
empty, will change to read "H = < x >," with x being whatever element you
added. This means that H is the subgroup generated by the
element you chose, not that H contains only the
element you chose. Second, all elements in the subgroup H
will be removed from the drop-down list for adding elements.
Removing elements from H: In the drop-down list next
to the "Remove this element from H" button, one will find a list of all
generators that have been added to H. Thus if the description
of H reads "H = < r, f >," then the drop-down list for removing
elements will contain two entries, r and f. To remove a generator,
select it in the drop-down list and click the "Remove this element from H"
button. Two changes will occur. First, the description of H
will change (e.g., to read "H = < r >" if you removed f). Second,
all elements that are consequently no longer in the subgroup H will
now be replaced in the "Add this element to H" drop-down list.
Here is one view of the second tab in the Edit Cayley Diagram dialog box.
Its instructions panel contains the following text (only part of which is
visible in the picture above). We add to these directions below.
Group Explorer can bring out parts of a group in its Cayley
Diagram, so the user might better analyze them. This page, the
"Highlight" page, is where that functionality is controlled.
Two different alterations in Cayley Diagrams can accentuate properties
of the group's elements:
1. the shape of the nodes corresponding to those group elements, and
2. the color of the nodes corresponding to those group elements.
To use node color to bring out properties of certain group elements,
select the desired property from the list on the left side of this
page.
- If the highlighting is of an on/off type (e.g. the group's
center--either an element is in it or it is not) then the elements
highlighted in yellow have the desired property (e.g. being in the
group's center) and those that remain unhighlighted (the usual dull
white color) do not have the property.
- If the highlighting is of a type that partitions the group into more
than two categories (e.g. each conjugacy class) then several colors
will be used to show that partitioning.
- In the specific case of highlighting each coset, the
yellow-highlighted elements are the subgroup, and all other colored
elements are in various cosets.
To use node shape to bring out properties of certain group elements,
select the desired property from the list on the right side of this
page.
- If the highlighting is of an on/off type (e.g. the group's
center--either an element is in it or it is not) then the elements
that remain spherical have the desired property (e.g. being in the
group's center) and those that are drawn as triangles do not have the
property.
- If the highlighting is of a type that partitions the group into more
than two categories (e.g. each conjugacy class) then several shapes
will be used to show that partitioning. This breaks down into two
cases:
1. If the partitioning can be described numerically (e.g. order
classes each correspond to an order) then the number corresponding to
each partition will be the number of sides used for its shapes. So the
elements of order 3 will be shown with triangles. Spheres are used to
correspond to the number 1.
2. If the partitioning cannot be described numerically, then the shape
assignment has no such interpretation. In this case, we avoid using
dashes (two-sided polygons) due to their distinctly different
appearance from all other polygons.
- In the specific case of highlighting each coset, the spherical
elements are the subgroup, and all other shaped elements are in
various cosets.
The "Ensure nodes and arrows use different colors" checkbox is used to
keep colors as distinct as possible. This option is only available if
the user has chosen to highlight an aspect of the diagram in color.
There are advantages and disadvantages to each option:
- If the box is checked, this means that the choosing of colors for
nodes does not pay attention to the choosing of colors for arrows, and
so you may have nodes that have the same color as arrows, even though
this is only coincidental.
- If the box is unchecked, this will ensure that nodes and arrows
choose distinct colors from one another. An advantage to this is that
it can avoid confusion that may arise from the assumption that like
colored things have something to do with one another. A disadvantage
is that more colors need to be used, and thus each color will be less
distinguishable from the others.
To turn off all highlighting, click the "Remove all highlighting"
button. To revert to the last highlighting configuration that
was approved via the "OK" or "Apply" buttons, click "Last
highlighting." Be aware that this may alter the contents of the
"Define H and a" page also, because H and a revert to their former
values as well. Neither of these takes effect until "OK" or
"Apply" is clicked. |
Examples
Because the above documentation that appears in the instructions panel of
the highlighting page is fairly comprehensive, we include here a few
examples of how highlighting can be useful.
Cosets - To see how cosets partition a group, color highlighting is
most useful. Try the following step-by-step process to highlight the
right cosets of a group, then try it again to highlight the left cosets.
| Use the Define H and a tab to
define a subgroup H whose cosets you wish to highlight. For
example, you might open the group S3 and highlight the
subgroup < f >. |
| Switch to this tab (Highlight) and choose the "each right coset Hg"
option from the "Use node color to highlight:" list. |
| Click OK or Apply to have your choices take effect in the diagram. |
The left and right cosets will coincide if the group is normal. To
read more about editing diagrams to visualize normality, see
the extended example at the end of
this page.
Class Equations - One can read off the class equation of a group by
choosing the "each conjugacy class" option from the "Use node color to
highlight" list.
For example, in S3, one would see one magenta sphere,
two yellow spheres, and three light blue spheres. This separates the
group into three conjugacy classes: one of size 1, one of size 2, and one
of size 3. Thus the class equation would read 1 + 2 + 3 = 6.
Order Classes - One can see the order of each element of a group by
choosing the "each order class" option from the "Use node shape to
highlight" list.
Using S3 as our example again, we would find three
different shapes appearing when we highlight order classes by node shape.
The shape each element takes on will have a number of sides equal to the
element's order. For example, two elements in the group will become
triangles, indicating that they have order three. Three elements in
the group will become dashes (-), indicating they have order two.
The identity element will remain a sphere, indicating that it has order 1.
Here is one view of the third tab in the Edit Cayley Diagram dialog box.
Its instructions panel contains the following text (only part of which is
visible in the picture above). We add to these directions below.
When performing auto-generation of Cayley Diagrams, Group
Explorer uses a set of generators for the group to build the diagram.
In addition to the set of generators, it is also necessary to assign
an axis and a priority to each.
This page allows the user to instruct Group Explorer regarding which
generators to use when auto-generating the diagram. The "Axes &
Priority" page addresses the assignments of axes and priorities to
those generators.
If the diagram being edited is not auto-generated, the options on this
page do not apply. In such a situation, the heading immediately below
this set of directions will be in bold, and will read "Changing
generators is not available for custom diagrams."
Important: From Group Explorer's Group menu, one can choose a set of
generators for the current group. These govern what appears in the
Navigator view, and what generators are used by default to
auto-generate Cayley Diagrams and to place arrows among their nodes.
However, the set of generators used in auto-generation of Cayley
Diagrams need not remain
equal to the set of generators used in the Navigator. This page allows
alteration of that set of generators independent of the generators
used in the Navigator, and also independent of which arrows are drawn
among the nodes in the Cayley Diagram.
When specifying generators for the group, it is essential that the set
of elements chosen actually generate the entire group, and are not
redundant. Two features are in place to ensure this occurs:
1. The selections you make on this page only affect the other pages in
this dialog if the list of generators generates the whole group.
2. If you attempt to apply changes to the diagram with this page in an
unacceptable state, the selections on this page revert to their last
valid configuration.
3. The list of elements available to add to the list of generators
only contains elements that are not already part of the subgroup
generated by those elements already in the generators list.
Note that it is still possible, even given these features, to make a
list of generators that are redundant, based on the order in which you
add them. You will get valid Cayley diagrams in such circumstances,
but they are usually not the most attractive, or are awkwardly
positioned.
To revert the list of generators to its default, which is the
currently selected set of generators for the group, as per Group
Explorer's Group menu, click the "Current group generators" button. To
revert the list of generators to the last set that was approved via
the "OK" button or the "Apply" button, click the "Last set of
generators" button.
Note: The "Organize diagram by H" button on the "Axes & Priority" page
may have impact on this page. Refer to the directions on that page for
more information. |
Generating a group well
It is worth noting that the controls on this page for adding and removing
generators behave very much like the controls on
the Define H and a tab for placing
generators into the subgroup H. The way one can check to be
sure they have generated the whole group with the generators list they
have constructed is by checking the drop-down list next to the "Add this
generator:" button. If there are choices in the drop-down list, then
there remain unreached elements of the group. If no choices remain,
the whole group is generated by the current list of generators.
Generating a group badly
The directions above that appear in the instructions pane on this tab
point out that it is possible to assemble a redundant set of generators
for the group. For example, in the group Z2 x
Z6, the generators <a,e> and <e,b> generate the
group. But the generators <a,e> and <e,bb> do not.
Therefore, if the user adds generators in the order <a,e>, <e,bb>, <e,b>,
Group Explorer will not complain, and will in fact be deceived into
thinking that these generators will create a three-dimensional Cayley
diagram. In reality, it will only be two-dimensional, and therefore
will look awkward when Group Explorer tries to graph it
three-dimensionally.
A better interface would probably fix this problem, but it is sufficiently
small an issue that other concerns were more pressing. Perhaps a
future version will improve the situation.
Here is one view of the fourth tab in the Edit Cayley Diagram dialog box.
Its instructions panel contains the following text (only part of which is
visible in the picture above). We add to these directions below.
Once Group Explorer has a set of generators from which to
build a Cayley Diagram, it needs to know in what order those
generators should be used, and to what axes they should be assigned.
Group Explorer makes somewhat sensible default choices for these
options when auto-generating a Cayley Diagram for the first time, but
this page allows the user to change those defaults.
When generating a Cayley Diagram from a set of generators, the
priority Group Explorer assigns those generators will govern which
ones have local significance and which ones have global significance.
Lower-priority generators have local effects, and therefore the
subgroups generated by them tend to remain in the same shape, and
their cosets appear noticeably throughout the diagram. Higher-priority
generators handle the global connectivity of smaller pieces of the
diagram that are connected by lower-priority generators. For
example, in a group with two generators, the lower-priority generator
will make little one-dimensional cosets that will be recognizable
throughout the diagram, while the higher-priority generator will
connect those recognizable cosets in whatever way the group's
definition specifies, which may or may not involve any easily
recognizable pattern for the observer.
An independent choice from generator priority is the assignment of
generators to axes. For example, in a two-dimensional group being
drawn as a rectangular diagram, which of the two arrows should point
horizontally (the x-axis) and which should point vertically (the
y-axis)? This choice is independent of the priority of the various
generators. It is usually better to have higher-order generators on
circular axes, so they are spaced far apart; order-two generators tend
to look strange on circular axes.
To manipulate these properties, click a priority or an axis assignment
in the white portion of the grid shown directly below these
instructions, and move it up or down by clicking the "Move Up" button
or the "Move Down" button to the right of the grid.
Two important shortcut controls are available on this page:
1. The "Organize diagram by H" button changes all the priorities in
the grid to ensure that the subgroup H (as per the "Define H and a"
page) is of lowest priority, and therefore the subgroup and its cosets
will partition the diagram in a recognizable way. It is important to
note that doing this may require alteration to the list of generators
for the diagram, and therefore may have impact not only on this page,
but also on the "Generators" page. This button will only be enabled if
there is a nonempty subgroup H, as defined on the "Define H and a"
page.
2. The "Change all axes to" button changes the set of entries that
appear in the "Axes" column of the grid to match the schema the user
has chosen in the drop-down list. For example, a rectangular diagram
can be reshaped into a hollow cylindrical diagram by choosing "Hollow
cylindrical" from the drop-down list next to the "Change all axes to"
button, and then clicking the button. Some reorganization of the
assignment of each axis to a generator may remain to be done.
The "Change current axis to" button is a future feature that is not
yet enabled.
The "Default node size" slider allows the user to choose how large the
spheres are for the nodes in the Cayley Diagram. Group Explorer may
modify node size at any time in order to ensure nodes do not overlap,
and do not completely swallow the arcs connecting them, but this
slider allows you to set the default size from which such adjustments
are made. |
Although the instructions for this page are fairly complete above, this is
one of the most complicated of the tabbed pages for this dialog box, so it
bears further examination. We illustrate the instructions above with
the following pictures and examples. If these examples seem
inadequate, simply experiment with Group Explorer to furnish yourself with
further ones.
Generator priority
The instructions above explain that lower priority generators have more
local significance in the diagram. But what does this look like?
The following pictures are an example. The diagram shown is a polar
Cayley diagram for the group A4. The order-2
generator has lower priority, and the order-3 generator has higher
priority.
Default Cayley diagram in polar configuration |
Only the order-2 generator drawn
(note the locality of its influence,
and the equality among the copies) |
Only the order-3 generator drawn
(note that it connects the local pieces
built by the order-2 generator,
and forms differently-shaped copies) |
In abelian groups, the priority is of no consequence. To see why,
read the page on the Fundamental Theorem of
Abelian Groups.
Generator axis assignment
This assignment is more superficial than the generator priority
assignment. The generator priorities determine the way the elements
are laid out in Group Explorer's brain before they are placed into space.
The assignment of axes to generators chooses how that arrangement will be
laid out in space.
The significance of this choice is often dependent on the axes with which
one is working. For instance, if the system of axes is rectangular (x
and y), then the change in the diagram is simply to flip it on its
side, as shown in the following pictures.
Rectangular diagram of S3 with axes assigned
this way: x-axis with r and y-axis with f |
Rectangular diagram of S3 with axes assigned
this way: x-axis with r and y-axis with f |
However, if the system of axes is, for example, polar, the choice of axes
can make the difference between a bad Cayley diagram and a good one.
In the following pictures, we see that assigning an order-2 generator to
the theta axis looks awkward. When choosing axes for the first
displaying of a Cayley diagram, Group Explorer tries to put higher-order
generators with rotational axes, as well as following a few other
heuristics that try to keep the diagram attractive.
Polar diagram of D5 with axes assigned
this way: theta-axis with r and r-axis with f |
Polar diagram of D5 with axes assigned
this way: theta-axis with f and r-axis with r |
Default node size
Although there is not a lot of call for it, the default node size can be
changed. Here are a few illustrations.
Default node size unchanged |
Smaller default node size
(but not as small as they could get) |
Larger default node size
(but not as large as they could get) |
Here is one view of the fifth tab in the Edit Cayley Diagram dialog box.
Its instructions panel contains the following text (only part of which is
visible in the picture above). We add to these directions below.
This page involves the cosmetic aspects of the Cayley
Diagram, in the sense that none of the options below relocate any
nodes.
Two types of chunking are available in Cayley Diagrams, and both are
to assist the viewer in considering cosets within the diagram as
chunks, or single units, when attempting to visualize a quotient
operation. The two operations are these:
1. Chunking right cosets of the subgroup H, as per the "Define H and
a" page. This option will put translucent gray zones around each right
coset of H, to aid the viewer in considering the cosets as single
units rather than as complex objects with internal structure and
detail. This operation is only available when the generators for H
have been prioritized lowest on the "Axes & Priority" page, so as to
make each coset a local region within the whole diagram, not spread
out unpredictably.
2. Aligning the heads of arrows between right cosets of H can make it
more clear that a quotient operation has taken place, and that the
resulting partitioned diagram forms a group.
This option is only available when H is a normal subgroup of the whole
group.
Immediately below this set of directions are two checkboxes, one for
each chunking option. If the option is available, the text above the
checkbox will say so. Otherwise, it will explain why that option is
not available, given the current group, choice of H, and settings in
earlier pages.
You can also customize the appearance of arrows within the diagram in
the following two ways:
1. Editing the list of visible arrows using the "Add this arrow" and
"Remove selected arrow" buttons. You can add or remove any type of
arrows you like from the diagram, independent of your choices of
generators on the "Generators" page. Beware that careless choices here
can create diagrams that are quite tangled. Also, removing arrows can
leave a diagram disconnected. Although this is not wrong (i.e. the
information is correct), it can be confusing or less useful (i.e.
there isn't enough information to make sense or be helpful).
2. Changing the default thickness of arrows. Group Explorer
automatically thickens arrows in proprotion to the size of the image,
but this slider provides a means to choose the base from which such
scaling is done. |
Choosing arrows
It is worth noting that the controls on this page for adding and removing
generators behave very much like the controls on
the Define H and a tab for placing
generators into the subgroup H. The difference is that one
can choose as many or as few arrows as one would like to include in the
diagram, without worrying about what subgroup they generate. That
is, redundant arrows are acceptable (to the point of having an arrow for
every element of the group!) and no arrows at all is acceptable.
Each of these situations is of little use, but occasionally redundant
connections in a diagram can add symmetry, as with some of the custom
diagrams for the quaternion group Q4.
Chunking error messages
Group Explorer always gives reasons when it cannot chunk cosets or arrows,
but due to space considerations, they are sometimes terse. For that
reason, we list each possible error message here and give a fuller
description, together with suggestions for how to alleviate the error.
| Coset chunking error message |
Description and corrective action |
| The diagram is organized by
the subgroup H. Chunking cosets is available |
This message is not an error, but rather indicates
that the diagram is correctly organized to allow coset chunking. |
| Diagram not organized by H.
Generator x has too high priority. Chunking cosets unavailable |
This message indicates that the diagram is not
organized by the subgroup H. The easiest way to remedy this
is using the "Organize diagram by H" button on the Axes & Priority
page. But more importantly, the idea behind a diagram being
organized by a subgroup is that all the subgroup's generators should
be prioritized very low, so that they have local effects. (See
above, under the Axes & Priority tab.)
To remedy this error, use the Axes & Priority page to decrease the
priority (higher numbers are lower priority) for the generator
mentioned. |
| The current group does not
respond well to current organization algorithm. Chunking unavailable |
The current algorithm for chunking cosets when the
user clicks the "Organize diagram by H" button on the Axes & Priority
page is not perfect. There are some circumstances it cannot
handle, although they are quite rare. (For example, organizing
by the group < b, c > in Unnamed group #2 of order 16.)
In this situation, you will need to accomplish the organization by
hand, if possible. This is something we would like to fix in a
future release. |
| Cannot chunk around the ring axis. Chunking unavailable |
As you can tell by seeing how Group Explorer chunks
cosets, it would be both difficult and confusing to chunk cosets if
the system of axes were, say, ring of rectangles, and each coset were
formed from one element from each rectangle. This error comes up
when the subgroup H is one-dimensional (cyclic). Thus the
chunks would also be one-dimensional tori, and would need to weave in
and out amongst each other. This would be complex to create and
to interpret. Thus chunking involving the ring axis is
disallowed.
To remedy this problem, reassign the axes so that the ring axis is
assigned to a higher-priority generator (lower numbers are higher
priority). |
| Cannot chunk around the w axis. Chunking unavailable |
This error also comes up only when the subgroup H
is one-dimensional (cyclic). The reason is analogous to that for
disallowing chunking around a ring axis (above). To remedy this
problem, reassign the axes so that the w axis is assigned to a
higher-priority generator (lower numbers are higher priority). |
| Can only chunk the w axis if
the x axis is also chunked. Chunking unavailable. |
This error comes up when the subgroup H is
two-dimensional, and says that chunking using the w axis is viable,
provided that the x axis is the other dimension H uses.
This is because the w axis is a fourth dimension, and the way it is
embedded in R3 is by overusing the x axis.
Thus if the w and x axes share H, rectangular solids can be
used to chunk the cosets. (Try it.) But if the x axis is
not involved, the same problem as appeared above, under "Cannot chunk
around the w axis" occurs here also. To remedy this problem,
reassign the x axis to have high enough priority to be a chunking
axis, or reassign the w axis to have low enough priority to not be a
chunking axis. (The two lowest-priority axes will be used for
chunking in the case when H is two-dimensional.) |
| Can only chunk the ring axis
if the theta axis is also chunked. Chunking unavailable |
This error comes up when the subgroup H is
two-dimensional, and is analogous to the above error involving w and
x. This is because the ring axis and the theta axis both share
the x-y plane. To remedy this problem, reassign the theta axis to
have high enough priority to be a chunking axis, or reassign the ring
axis to have low enough priority to not be a chunking axis. (The
two lowest-priority axes will be used for chunking in the case when
H is two-dimensional.) |
| Can only chunk the ring axis
if the x and y axes are also chunked. Chunking unavailable |
This error comes up when the subgroup H is
three-dimensional, and is analogous to the above error involving w and
x. This is because the ring axis uses the x-y plane. To remedy
this problem, reassign the x and y axes to have high enough priorities
to be chunking axes, or reassign the ring axis to have low enough
priority to not be a chunking axis. (The three lowest-priority
axes will be used for chunking in the case when H is
three-dimensional.) |
| Can only chunk the w axis if
the r and theta axes are also chunked. Chunking unavailable |
This error comes up when the subgroup H is
three-dimensional, and is analogous to the above error involving w and
x. This is because the w axis is embedded using the x axis, and
both the r and theta axes make use of the x-y plane. To remedy this
problem, reassign the r and theta axes to have high enough priorities
to be chunking axes, or reassign the w axis to have low enough
priority to not be a chunking axis. (The three lowest-priority
axes will be used for chunking in the case when H is
three-dimensional.) |
| Can only chunk the ring axis
if the r and theta axes are also chunked. Chunking unavailable |
This error comes up when the subgroup H is
three-dimensional, and is analogous to the above error involving w and
x. This is because the ring axis uses the x-y plane, which the r
and theta axes also use. To remedy this problem, reassign the r and
theta axes to have high enough priorities to be chunking axes, or
reassign the ring axis to have low enough priority to not be a
chunking axis. (The three lowest-priority axes will be used for
chunking in the case when H is three-dimensional.) |
| No subgroup H defined.
Chunking unavailable |
This error comes up because
the Define H and a tab does not
contain a definition of H. Or rather, it defines H
to be empty. To remedy this, return to that tab and add some
elements to H. |
| |
|
| Arrow chunking error message |
Description and corrective action |
| The diagram is organized by
the subgroup H. Chunking arrows is available |
This message is not an error, but rather indicates
that the diagram is correctly organized to allow coset chunking. |
| Aligning arrowheads not
available because right cosets of H are not chunked |
This error message indicates that a separate error
message for coset chunking is present, and that because coset chunking
cannot occur due to that error, arrowhead alignment is therefore also
unavailable. To fix this problem, address the error for coset
chunking, as per the above table discussing those error messages. |
| Arrowheads can be aligned if coset
chunking is enabled |
It is only possible to align the arrowheads between
cosets if the cosets have already been chunked. To remedy this
problem and enable arrowhead alignment (chunking), simply click the
checkbox to chunk the cosets of H. |
| Arrowheads can only be
aligned when the subgroup H is normal. Alignment unavailable |
This message indicates that the current subgroup H
will never be able to be used for arrowhead alignment. This is
because arrowhead alignment is to be used for accentuating the
agreement among arrows from one right coset to another. That is,
all arrows from one right coset must go completely to another right
coset. Because the arrows in the diagram are arrows of left
multiplication, this only occurs when the subgroup is normal. Thus
when this does not happen, the arrows that leave one right coset are
not unanimous on their destinations; each may head in a different
direction. For this reason, aligning their arrowheads is not
possible.
For a better description of Cayley diagrams and normality, see
the Example section immediately
below, or the Illustrated Proof page
on Normality and Quotients. |
This section shows how to exhibit visually the normal subgroup in A4,
using Cayley diagrams. We begin by walking through all the steps
that manipulate an ordinary Cayley diagram of A4 into a
more useful diagram for this purpose. To accomplish this, we use
many of the tools discussed above on this page (the Edit Cayley Diagram
dialog box). We follow up that walk-through by examining the diagram
we've created and noticing what it is about the diagram that indicates the
normality of the subgroup. In fact, we'll even discover how we can
see the quotient operation before our eyes, and the resulting factor group
as well.
Exhibiting a normal subgroup in A4
We go about exposing the normal subgroup within A4 in
six steps.
-
Open a rectangular Cayley diagram for A4. It
is important to have an auto-generated diagram, because several
important options are not available for editing with a custom
diagram--all those that relate to how it is laid out in space.
Your diagram will look like this one.
This diagram is one that demonstrates Group Explorer's lack of
aesthetics. Although Group Explorer knows several heuristics to
often make decent Cayley diagrams, it does not always succeed.
So we'll want to manipulate this diagram to reveal some structure in
it, because right now it looks pretty unpleasant. We will assume
the knowledge that a particular subgroup of A4 is
normal and isomorphic to V4, and we will reorganize
the diagram to pick out that subgroup.
-
Using the "Define H and a" tab of the Edit Cayley Diagram dialog box
on the diagram, add the elements (0 1)(2 3) and (0
2)(1 3) to the subgroup H. These are two different
order-two elements that commute, and therefore the subgroup they
generate is isomorphic to the Klein-4 group V4.
When you have done so, the portion of your Edit Cayley Diagram window
that corresponds to the definition of H will look like this.
-
Now switch over to the Highlight tab so that we can make use of the
subgroup H we've defined. Under "Use node color to
highlight," choose "each right coset Hg," as shown here.
When you click OK or Apply, your diagram should look like this one,
which partitions the set of nodes into three distinct classes by
color, representing the right costs Hg of H, for all
g in A4.
This image shows us how the group is partitioned by the right
cosets of H, but it's still a fairly ugly picture. Let's
rearrange it to group together things of like color so we can see if
there are any patterns.
-
Group Explorer provides a functionality for bringing together the
cosets of the subgroup H defined on the first tab of the dialog
box. Find the Axes & Priority tab, and click the "Organize
diagram by H" button. Your grid of generators, priorities, and
axes, should then look like this one.
When you click OK or Apply, it will then modify your diagram to this
three-dimensional shape.
It was Group Explorer's best guess that this arrangement of
generators, priorities, and axes would be the best suited for the eye.
However, things are still a bit difficult to see here, even though
some more order is apparent. In fact, if we spin this shape a
bit to the right, we can line up in our view the yellow, blue, and
green nodes in columns. But the arrows are still a bit hard to
follow.
-
So let's choose a different assignment of axes. On the Axes &
Priority tab of the Edit Cayley Diagram box, go to the drop-down list
next to the "Change all axes to:" button and choose the "Ring of
rectangles" item. Click the "Change all axes to:" button to have
this choice impact the axes in the grid. Your grid should then
look like this one.
When you click OK or Apply, your diagram will then look like this.
This is a little better, in that it's localized the colors better, but
the arrows are still hard to follow. They all seem to be going
clockwise, but that's all that's apparent. Two key tools remain
to us, however, as we seek to investigate the subgroup V4
by which we've organized this diagram. They are the chunking
tools.
Any subgroup H can reorganize a diagram like we have done so
far with this particular H. But now we wish to
investigate whether the subgroup H is normal. We know
that H is normal if and only if we can use it as the
denominator to form a factor group (see
an illustrated proof of this). The chunking tools help us to
visualize such a factor group, if it exists. So let's use those
tools to see the factor group A4 / H.
-
On the Arrows & Chunking tab of the Edit Cayley Diagram window, check
the box for "Chunk right cosets of H" and then check the box for
"Align heads of arrows between right cosets of H" as shown in this
illustration.
Your diagram will then look like this one, and at this point we're
done manipulating the diagram.
You can see that Group Explorer has done two things here. First,
it put translucent gray boxes around the cosets of H so that
you can more easily think of them as single units. Thinking of
cosets as individual items is the first step in forming a factor
group. Second, it brought the arrowheads on the red arrows near
to one another, so that they group nicely. This demonstrates the
agreement of the red generator about how to operate on the cosets.
We discuss this further below.
Understanding the result
What is the significance of this final picture? What can it tell us?
It tells us about the normality of V4 in A4,
and it shows us the factor group one gets when taking the quotient.
Let's see how.
| One can see that the red arrows that leave the top coset (the yellow
four nodes) all move unanimously to the right (blue) coset. That
is, the red arrows take the four yellow nodes to the four blue nodes.
Similarly, they take the four blue nodes to the four green nodes, and
then the four green nodes back to the four yellow nodes. The
reason this tells us that H is normal is that we're viewing
right cosets of H, and the Cayley diagram is generated by
left multiplication. Yet this diagram shows us that those
things coincide. That is, the red arrows tell us what left
multiplication by the element (1 3 2) does, and so
following four red arrows from four yellow nodes ought to arrive us at a
left coset of H, but we find it arrives us at the four
blue nodes, a right coset of H. Thus left and right
cosets coincide for H, making it normal. |
| Furthermore, one can see the factor group A4 /
V4 by taking a few steps back from the screen. That
is, go ahead and picture the big gray boxes as individual elements,
because after all, that's what happens in a factor group--cosets in A4
become elements in A4 / V4.
Then notice that among the three elements in the factor group (the three
gray boxes) we have one generator, represented by red arrows.
Those arrows begin at the top and march clockwise through the three
elements, in cyclic fashion. Thus the factor group is isomorphic
to the cyclic group Z3. Consider the
following amateur illustration of the isomorphism. |
For additional discussion of normal subgroups in Cayley diagrams, see
the Illustrated Proof page on Normality and
Quotients.
|
