Editing Multiplication Tables
Note: This page has a twin at Editing
Cayley Diagrams.
Outline
The purpose of this page is twofold. First, we give a detailed
account of the uses of the Edit Multiplication Table dialog box. The Edit
Multiplication Table dialog box is a window with four tabbed sheets, each of
which has several controls on it. For this reason, it has
significant complexity, which can be initially frustrating. 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 color to visually
indicate properties of some elements in the group |
| Tab 3: "Generators" - For changing the
list of group elements used as generators for the table |
| Tab 4: "Priority &
Chunking" - For assigning priorities to each generator, and
for visually separating the cosets of H |
| Example - Exhibiting normal subgroups visually by editing
the multiplication table |
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.
Because this tab is identical to the first tab in the
Edit Cayley Diagram dialog box, we do
not duplicate the documentation. Rather, we refer the reader to
that documentation.
Here is one view of the second tab in the Edit Multiplication Table 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
Multiplication Table, so the user might better analyze them. This
page, the "Highlight" page, is where that functionality is controlled.
Two different alterations in Multiplication Tables can accentuate
properties of the group's elements:
1. colored wedges by the names of the group elements in the leftmost
column (row headings), and
2. the background colors of the elements within the table.
To use coloring of the left column to bring out properties of certain
group elements, select the desired property from the list on the left
side of this page.
To use coloring within the table 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
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 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
On the Editing Cayley Diagrams page,
we discuss three examples of the usefulness of highlighting in Cayley
diagrams. Only two of those examples remain valid in this medium.
Because the Multiplication Table does not contain a means to attach a
number to a group element (as we did with shapes in Cayley diagrams) we
cannot discern immediately the order of an element by highlighting order
classes by shape, as we did in Cayley diagrams. However, the
following interesting examples remain.
Cosets - To see how cosets partition a group, color highlighting
within the table 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 entire table to highlight:" list. |
| Click OK or Apply to have your choices take effect in the table. |
The left and right cosets will coincide if the group is normal. To
read more about editing the table 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
left column to
highlight" list.
For example, in S3, one would see one element
highlighted in yellow,
two elements highlighted in light blue, and three elements highlighted in
magenta. 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.
Here is one view of the second tab in the Edit Multiplication Table 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 Group Explorer builds a Multiplication Table, it uses a
set of generators for the group. In addition to the set of
generators, it is also necessary to assign a priority to each.
This page allows the user to instruct Group Explorer regarding which
generators to use when generating the table. The "Priority & Chunking"
page addresses the assignments of priorities to those generators.
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 generate
the Multiplication Table. However, the set of generators used in
generation of the Multiplication Table 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.
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 table 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.
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 "Priority & Chunking"
page may have impact on this page. Refer to the directions on that
page for more information. |
Choosing a sufficient list of generators
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.
The Editing Cayley Diagrams page
warns you not to add generators in an order that causes the list of
generators to be redundant, because this can fool Group Explorer into
making less-than-optimal Cayley diagrams. However, there is no such
problem with multiplication tables. Although you can create
redundant lists of generators, this only affects the order of the elements
as they are listed across the top and down the left side of the
multiplication table. Thus no awkwardness results.
Here is one view of the second tab in the Edit Multiplication Table 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 the Multiplication Table, it needs to know in what order those
generators should be used. Group Explorer makes default choices for
these options when generating the Multiplication Table for the first
time, but this page allows the user to change those defaults.
When generating the Multiplication Table 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 form recognizable boxes in the
upper-left corner of the table, and their right cosets appear
noticeably across the top of the table.
To manipulate generator priorities, click a priority 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.
The "Organize table 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 table in a recognizable way. It is important to note
that doing this may require alteration to the list of generators for
the table, 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. |
Generator priority
The instructions above explain that lower priority generators have more
local significance in the table. But what does this look like?
The following pictures furnish us with an example. The table shown
is for the frobenius group Fr20.
Multiplication table for Fr20
with priority assigned to generator s |
Multiplication table for Fr20
with priority assigned to generator t |
In the above tables, you can see quite a structural difference. The
table on the left begins listing the elements of the group across the
column headings with the powers of the generator s, and when it
runs out of elements of the form sk, then it begins
using the generator t. Thus that table is organized by the
subgroup generated by s, a subgroup which is not normal in Fr20.
The diagram on the right has the roles of s and t reversed,
and because the subgroup generated by t in Fr20
is normal, you can see the factor group in the overall structure of the
multiplication table. If you are not familiar with visualizing
factor groups in multiplication tables, read
the example section on that topic,
below.
Chunking right cosets of H
Chunking cosets of a normal subgroup makes visualizing the factor group
easier. The following two illustrations demonstrate the difference
chunking makes. Chunking simply adds blank space between the cosets
of H so they are easier to separate visually. For more on normal subgroups in multiplication
tables, read the example section on
that topic, below.
Multiplication table for Fr20
organized by the subgroup < t >
with coset chunking disabled |
Multiplication table for Fr20
organized by the subgroup < t >
with coset chunking enabled |
Chunking error messages
Group Explorer always gives reasons when it cannot chunk cosets,
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.
Compared to chunking cosets and arrows in a Cayley diagram, this situation
is fairly simple.
| Chunking error message |
Description and corrective action |
| The table is organized by
the subgroup H. Chunking cosets is available |
This message is not an error, but rather indicates
that the table is correctly organized to allow chunking. |
| Table not organized by H.
Generator x has too high priority. Chunking cosets unavailable |
This message indicates that the table is not
organized by the subgroup H. The easiest way to remedy
this is using the "Organize diagram by H" button on this
page. But more importantly, the idea behind a table being
organized by a subgroup is that all the subgroup's generators should
be prioritized very low, so that they have local effects, as discussed
above.
To remedy this error, use the "Organize table by H" button, or
manually decrease the
priority (higher numbers are lower priority) for the generator
mentioned. |
| 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. |
This section shows how to exhibit visually the normal subgroup in Fr20,
using the Multiplication Table. We begin by walking through the
steps that manipulate the default Multiplication Table for Fr20 into a
more useful one for this purpose. To accomplish this, we use
many of the tools discussed above on this page (the Edit Multiplication
Table
dialog box). We follow up that walk-through by examining the table
we've created and noticing what it is about the table 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 Fr20
We go about exposing the normal subgroup in Fr20 in five
steps.
- Open Fr20 and view its multiplication table.
The default table Group Explorer presents looks like the one pictured
below.
You can uncheck the "Highlight current element" checkbox, since we
won't need that feature right now and it sometimes gets in the way.
- Using the Edit Multiplication Table dialog box, under the "Define H
and a" tab, add the element t to the subgroup H.
The following image shows what your definition of H should then
look like.
Clicking OK or Apply at this point will not alter your table at all,
because we have only defined H, and not actually done anything
with it yet.
- On the "Priority & Chunking" page, click "Organize table by H."
This instructs Group Explorer to make the generator(s) of H have
as low a priority as possible, so that the table will have the cosets of
H grouped together in little local clumps. After you click
"Organize table by H," your priority grid should look like the one in
the following illustration.
Clicking OK or Apply at this point will rearrange the order of the
elements in the table so that the cosets of H are near each
other, forming little local 5-by-5 clumps that can be distinguished by
color. Here is our multiplication table after organizing by H.
- On that same page, click "Chunk right cosets of H," as shown
here.
This instructs Group Explorer to make some extra space between the
5-by-5 cosets of H, so that you can more easily group them
together visually.
Our diagram is now (nearly) perfectly suited to seeing the normality of
the subgroup H, and the factor group of Fr20 by
H. In fact, such things are already becoming apparent.
Let's solidify this chunking scheme further with a uniform coloration.
- On the "Highlight" page, under the "Use entire table to highlight"
heading, choose "each right coset Hg."
Click OK or Apply to
perform the highlighting, and your table should look like this.
In the previous picture, each 5-by-5 square had its own 5-color
palette, but now we make each such 5-color palette a uniform color.
That is, the upper left used to contain reds and oranges, and there were
three other such 5-by-5 squares throughout the table; now those have all
become yellow. Similarly, the upper right used to contain violets
and magentas, and that pattern marched all the way down and to the left;
now all such squares are colored uniformly purple. And so on.
We now point out how this illustrates the quotient operation and the
factor group.
Understanding the result
Much as with Cayley diagrams, once the picture is rearranged to best
illustrate the normality of the subgroup, the thing to do is take several
big steps away from the screen. If we do so, we begin to see each
5-by-5 square in the above grid as a single square of a single color.
These larger squares then form a large 4-by-4 grid, which can be seen as a
multiplication table in its own right. That is, treating the cosets
as single units (which is how one forms a factor group), we find that the
resulting table is a valid group multiplication table.
This is due to the normality of the subgroup. Had the subgroup not
been normal, we would not have had the nice segregation of colors that
showed up earlier, in order to uniformize those colors like in the above
image. (Try repeating the experiment using the subgroup generated by
s in place of the subgroup generated by t.)
So we can see 4-by-4 the multiplication table of the factor group here,
but what group is it? It is the cyclic group Z4.
Consider the following illustration, in which colors exhibit the factor
group homomorphism from Fr20 on the left to Z4
on the right.
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