APPENDIX: GROUP IDENTIFIERS IN GALOIS
The program includes extensive resources for reviewing, analyzing, and comparing the groups that have been generated, either in the current GALOIS session or in a previous session.
*After Évariste Galois (1811-1832), the brilliant young French mathematician who introduced the concept of groups.
The program includes a Main Screen from which a hierarchy of secondary panels can be invoked. At any time, the user can obtain Help for any screen or panel by pressing F1, save a copy of the screen or panel (with buttons and other interactive controls suppressed) by pressing F2, exit from the screen by pressing F3, or terminate the entire application by pressing F12.
Some of the displays generated by GALOIS may exceed the dimensions of the monitor. In these screens, the cursor appears as a hand icon, which may be used to scroll up, down, left, and right. The Scrolling menu can be used to undo the effects of scrolling or to disable the scrolling feature.
The core of the Main Screen is a Cayley table presenting the elements and products that define the group. By default,* the order of the group is 6, but the order can be changed to any value from 1 to 15 by selecting Change Group Order from the Options menu. The group's identity element is denoted by the numeral 1, and the remaining elements by lowercase letters (a, b, c, ...). These element names (which can be modified through the Element Names Panel) identify the rows and columns in the displayed table. The products of the elements are represented by textboxes. Since x . 1 = 1 . x = 1 for any x in the group, the values in the first row and in the first column are predetermined and cannot be modified. The user can enter values into some or all of the remaining textboxes. The values must themselves be valid element names, since every group is by definition closed. After entering some values, the user can click Check to have the program check for errors, or alternatively the program can be used to search automatically for a solution.
*This default and several others can be modified by the user through the Preferences Panel.
Check Button
If the user enters values and then clicks Check, GALOIS checks whether any required properties of groups have so far been violated, and also whether any additional table values can be inferred based on properties of groups. For example, since every element in the group must have an inverse, xy = xz is only possible if y = z. (If xy = xz, then x-1xy = x-1xz and therefore y = z.) Thus the same element cannot appear twice in the same row of the table, and consequently each row must contain all of the group's elements in some sequence. By similar reasoning, each element must appear once and only once in each column. If any of these conditions are violated, a corresponding error message is returned.
An error message is also returned if the values entered violate the requirement that the operation be associative. Finally, an error message may be returned if the values entered are incompatible with other known properties of groups, even though those properties may not be explicitly named in the definition of a group. The properties to be applied can be controlled by the user (see the Optimizing Rules Panel).
Table values that have been inferred by the program are indicated in red. If the mouse is placed over one of these values, a message appears explaining briefly how the value was inferred. Instead of typing these values, the user can click on them to incorporate them into the table.
If the user has completely filled the table and no errors were found, a Solution Analysis Panel (discussed below) is automatically displayed.
Get Next Step / Get Next Solution Button
This button directs GALOIS to search automatically for valid group structures. (If values have previously been entered manually, they must first be cleared from the screen by clicking the Restart button.) In an automatic search GALOIS tests possible table values in a right-to-left, top-to-bottom sequence. The caption on the button depends on the Trace Option (see Trace Options below) selected by the user. By default (unless overridden by the Preferences Panel), the search uses a "Very fast trace" mode, but initially the user may wish to view it in slow motion by changing the option to "Trace by step." In step mode, the button is labeled "Get Next Step." By repeatedly clicking the button (or pressing ENTER), the user can observe the search process inferring new values (shown in red) from old, then adding new values to the remaining blank slots, or removing a value and backing up after a violation of the group requirements has been discovered. When a complete solution is found, it is displayed with analysis in the Solution Analysis panel. After the user has viewed this panel and clicked OK, the search can resume.
If the user wishes to view the search more rapidly, the Trace can be set to a different option or turned off altogether, as described in Trace Options, and the label on the button changes to "Get Next Solution." When no more solutions can be found, a message is returned asking whether the user wishes to create a Summary View file.
Include Isomorphic Equivalents Checkbox
Many of the possible solutions are isomorphically equivalent. Consider, for example, the following two Cayley tables:
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Consider the one-to-one mapping f from the set {1,a,b,c} onto itself, where f(1) = 1, f(a) = b, f(b) = a, and f(c) = c. Applying this mapping to Table X gives Table X' (shown in the middle):
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Upon closer inspection, if Table X' is rearranged to display the elements in their usual order (by exchanging the a-column and b-column, and then the a-row and b-row), it is seen to be equivalent to Table Y. The mapping f therefore establishes an isomorphism between the two groups.
By default (unless overridden by the Preferences Panel), solutions that are isomorphically equivalent to earlier solutions are excluded from the solution search in this program, reducing the number of possible solutions and also enabling the search algorithm to eliminate many possibilities and therefore to run much more rapidly. By checking the Include Isomorphic Equivalents checkbox, however, the user can view the whole range of solutions; if the group order is 4, for instance, both Table X and Table Y will be returned as solutions.
Restart Button
If the user has already entered values into some of the textboxes before initiating the search, then earlier solutions may not be found by the search. Clicking the Restart button avoids this problem by clearing all the textboxes (except for the first row and first column, which are predetermined), thus ensuring that the whole range of possible solutions will be found. Changing the Group Order also clears the textboxes in preparation for a subsequent search.
The Trace Option can be selected by the user from the Options menu. The available options are as follows:
The order of the desired solution groups can be changed by the user using the Options menu. The Group Order must be numeric and in the range 1-15. When it is changed, the textboxes are automatically cleared and the search for solution groups is reset to the beginning.
When no more solutions can be found, the user is given an opportunity to save the results in a Summary file in html format. If a Summary file with the chosen filename already exists, the results will be appended following the previous results; otherwise, a new file is created. This Summary is then displayed in a pop-up browser window.
The Summary View may include any or all of the following components for each solution group:
When a valid group has been found from the Main Screen, the resulting solution is automatically displayed in a Solution Analysis panel. The analysis first determines the order of each element in the group and uses that information to derive a set of generators for that group. The generators are designated using uppercase letters, in order to distinguish them from the lowercase letters used for elements on the Main Screen. Typically, the elements of a group can be described succinctly in terms of a small number (perhaps one to three) generators. A "reductive mapping" is provided, defining each of the original elements in terms of the generators. For example, if the solution is a cyclic group of order 6, then the mapping will relabel the elements as 1, A, A2, A3, A4, and A5, not necessarily in that order. The Cayley table is displayed using these revised element names.
At the bottom of the panel, specific properties of this group are analyzed, indicating whether or not the group is abelian, cyclic, dihedral, and/or solvable. Also noted are significant isomorphisms between the group and others such as symmetric groups, alternating groups, and additive or multiplicative integer modulo groups. In addition, every group is isomorphic to the permutation group defined by its rows and to the permutation group defined by its columns. (This fact gives rise to Cayley's Theorem. If the group is abelian, the rows are identical to the columns, so the two permutation groups are one and the same.) For each isomorphism, a View Group button and a Show Isomorphism button are displayed. By clicking the View Group button, the user can create and view a View File for the isomorphic group in html format. (For details, see the Group Analyzer Panels associated with these special groups.) The Show Isomorphism button opens the Isomorphism Panel, where a isomorphic relationship between the two groups is exhibited in detail.
If the Include Isomorphic Equivalents checkbox was checked on the Main Screen, then the current solution may be isomorphically equivalent to a previously found solution. If so, then that equivalence is noted at the top of the Solution Analysis Panel and can be examined further by clicking another Show Isomorphism button.
For relatively simple groups, such as cyclic and dihedral groups, the Solution Analysis includes a Show Graph button, giving access to a Graphic Representation Panel showing the group structure in visual form. In addition, a Show Subgroups button enables the user to view all the group's subgroups in a Subgroup Panel, and a Show Composition Series button opens the Composition Series Panel.
Afer viewing the Solution Analysis Panel, the user can return to the main menu by clicking OK or by pressing F3, after which a search for further solutions can be resumed.
The Subgroup Panel is displayed when the user clicks the Show Subgroups button from the Solution Analysis Panel. It can also be invoked from the Analysis Tools menu on the Main Screen. If opened from the Solution Analysis Panel, the Subgroup Panel shows all the subgroups of the solution group from largest to smallest, providing each subgroup's order, indicating whether or not it is a normal subgroup, and listing its elements. For each normal subgroup, a Show Quotient Group button gives access to the Quotient Group Panel, where the quotient group corresponding to this subgroup can be seen.
At the bottom of the panel, the center of the group and the commutator subgroup are also identified, and two calculators are provided for identifying centralizer and normalizer subgroups. The Centralizer Subgroup Calculator can be used to determine the centralizer of any element of the group. The user selects the desired element from a listbox, and GALOIS returns the Group ID of the corresponding centralizer subgroup, which was listed with its elements earlier on the same panel. Similarly, the Normalizer Subgroup calculator determines the normalizer of any of the subgroups. When the user selects a subgroup from a listbox, GALOIS returns the Group ID of that subgroup's normalizer subgroup.
If this panel is invoked from the Analysis Tools menu on the Main Screen, a Choose Group button is first displayed, enabling the user to select a group from the displayed Generated Group List. Because of limitations on window sizes, this panel can accommodate a maximum of 60 subgroups. If the group has more than 60 subgroups, the listing is identified as a "partial list" and only the first 60 subgroups that were found are displayed.
After reviewing this panel, the user can return to the previous panel by clicking OK or by pressing F3.
The Subgroup Conjugacy Class Panel is reached by clicking Show Subgroup Conjugacy Classes on the Subgroup Panel. It lists the subgroup conjugacy classes for a given group, along with their constituent subgroups. (This panel is similar in concept to the Conjugacy Class Panel, except that the classes in the latter consist of conjugate elements, while the classes on this panel consist of conjugate subgroups.) The subgroup conjugacy classes are identified one by one and the subgroups in each class are listed. Each subgroup conjugacy class Cl(H) consists of all those subgroups that are conjugate to a particular subgroup H. Since a subgroup is conjugate to H if and only if it can be written as x-1Hx for some element x, the subgroup is also expressed in this form.
At the bottom of the screen, an equation is worked out for the total number of subgroups, based on the number of subgroups in each subgroup conjugacy class. Because this equation uses the orders of normalizer subgroups, the user may also wish to refer back to the Subgroup Panel, which includes a calculator for normalizer subgroups. After reviewing all the information on the Subgroup Conjugacy Class Panel, the user can clicks OK to return to the Subgroup Panel.
The Quotient Group panel is displayed when the user clicks the Show Quotient Group button next to a normal subgroup on the Subgroup Panel. The order of the quotient group is given, the elements in the normal subgroup and in each of its other cosets are listed, and the Cayley table for the quotient group is displayed. After reviewing the quotient group, the user returns to the Subgroup Panel by clicking OK or by pressing F3.
The Graphic Representation Panel is displayed when the user clicks the Show Graph button from the Solution Analysis Panel. This panel displays a graph which presents the structure of the group in a concise and visually compelling manner. Each element is represented by a node, and the effect of multiplying the elements by each of the generators is represented by arrows connecting the nodes. From the graph, any product in the group's Cayley table can be easily determined. For example, if AB and A2B are two of the elements, then the element represented by AB . A2B can be determined by beginning at node AB and following the path determined by the three arrows labeled A, A, and B in that sequence, thereby arriving at the node representing the product.
As with the other screens in the system, an image of the graph can be saved to the clipboard by pressing F2, after which it can be pasted into other applications as desired. After viewing this panel, the user can return to the Solution Analysis Panel by clicking OK or by pressing F3.
The Composition Series Panel is displayed when the user clicks the Show Composition Series button from the Solution Analysis Panel. It can also be invoked from the Analysis Tools menu on the Main Screen. If opened from the Solution Analysis Panel, this panel identifies the normal subgroup at each stage of the group's composition series, listing its elements, giving its order, calculating the composition factor for that step, and determining whether or not the latter is prime. Based on this analysis, the group is classified as solvable or unsolvable.
If this panel is invoked from the Analysis Tools menu on the Main Screen, a Choose Group button is first displayed, enabling the user to select a group from the displayed Generated Group List. The Composition Series for that group is then displayed.
The Isomorphism Panel is invoked when the user clicks the Show Isomorphism button next to one of the groups identified as isomorphic to the solution group on the Solution Analysis Panel, or the same button displayed when an isomorphism is found by the Group Comparison Panel. If two groups are isomorphic, there must exist at least one function f that establishes a one-to-one correspondence between the two groups, such that the fundamental group structure is preserved by the function. Specifically, if the first group's operation is labeled . and the second group's operation is ~, then f(x)~f(y) must equal f(x . y) for any x, y in the first group. This panel identifies such a function f by indicating the specific element in the solution group to which each element in the other group is mapped. The two Cayley tables are displayed side by side so that the equality between each f(x . y) and the corresponding f(x)~f(y) can be readily verified.
The Cayley tables on this screen can be displayed either in conventional Row and Column Format or in a more compact Calculator Format, and elements can be represented either by their original names or their reductive-mapping names. If Calculator Format is used, then the product of two elements can be determined by clicking on those elements on two drop-down lists. By default, large tables are displayed in Calculator Format and smaller ones in Row and Column Format, while the elements are represented by their original names. These defaults can be overridden by means of the Table Format menu (or permanently through the Preferences Panel).
When groups are entered manually, it may sometimes be more convenient to replace the generic element names (1, a, b, c, ...) with names that are more meaningful for particular kinds of analysis. This Element Names Panel, which is accessed from the Options menu on the Main Screen, enables the user to substitute element names of his or her choosing. For example, in constructing the group of symmetries of an equilateral triangle (i. e., the dihedral group of order 6), the three rotations and three reflections might be assigned mnemonic names as in the table below.
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This panel can be invoked again at any point during the manual entry of groups, and elements that have already been entered will be translated to the new names as needed. The panel also features a Revert to Default Names button, which returns the element names to their default values (1, a, b, c, ...).
The modified element names entered on this panel apply only to manually entered groups. Groups found through automatic search always use the default names 1, a, b, c, ....
The Optimizing Rules Panel is accessed from the Options menu on the Main Screen. In order to make the search process more efficient by eliminating many possible combinations long before the table is filled, GALOIS can detect whether the values at any step in the search may be incompatible with certain known properties of groups, even though those properties may not be explicitly named in the definition of a group. For example, from Lagrange's theorem it is known that the order of a finite group must be divisible by the order of an each of its elements. Therefore, if a group of order 10 is sought and inserting a particular value in the product table would cause one element to have an order of 3, then that value can be eliminated from consideration. Similarly, if it can be shown (based on known group properties) that certain configurations of the table can only lead to solutions that are isomorphic to previous solutions, then those configurations can be eliminated if the Include Isomorphic Equivalents Checkbox is not checked.
The specific group properties used to optimize the search are listed on the Optimizing Rules Panel. The user has the option of deselecting particular properties on this panel in order to observe how they affect the search process. Updates to this screen are effective only for the current GALOIS session, and they require a restart to any search for groups that may be in progress on the Main Screen. When the user clicks OK, the program issues a warning that any search will be restarted; if a restart is not desired, the user can cancel changes and click Exit to leave the panel.
The Special Group Panels are reached through the Special Groups menu on the Main Screen. Unlike the groups generated by searching from the Main Screen, the groups generated through these panels have a specific predetermined structure. Their elements typically are well-defined mathematical objects, in contrast to the generic elements (1, a, b, etc.) in the groups found through the Main Screen. While the search operation on the Main Screen is limited to groups with orders 1 through 15, a special group may contain as many as 120 elements. Because of the potential large size of these groups' Cayley tables, the Special Groups are presented and analyzed in external View Files in html format. Alternatively, the user can generate a special group without creating an html View and then use one or more of the Analysis Tool Panels to examine and/or analyze the group. Each of the generated groups is referenced by a Group ID, assigned by GALOIS. These Group IDs are described in the Appendix.
The Special Group Panels include three panels for generating permutation groups, two panels for generating groups of integer congruence classes using modulo addition and multiplication, and panels for generating direct product groups and automorphism groups:
The Symmetric Group Analyzer Panel is accessed from the Special Groups menu on the Main Screen and enables the user to generate symmetric groups directly, rather than through a search process. The user types in the degree of the desired symmetric group and clicks Generate Symmetric Group. If the degree is d, the order of the resulting symmetric group Sd is d!, and the resulting Cayley table has d!2 entries. Because the sizes of these groups and tables increase so quickly, GALOIS only accommodates degrees in the range 1-5.
If requested, the display of the generated group is stored in a Symmetric Group View File in html format. If a file with the chosen filename already exists, the output will be appended following previous output; otherwise, a new file is created. This View is then displayed in a pop-up browser window.
The Symmetric Group View may include any or all of the following components:
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A Symmetric Group View will also be generated if the symmetric group is isomorphic to a solution analyzed on the Solution Analysis Panel, if the user clicks that View Group button.
The Alternating Group Analyzer Panel is accessed from the Special Groups menu on the Main Screen and enables the user to generate alternating groups directly, rather than through a search process. It is similar to the Symmetric Group Analyzer Panel, except that the generated alternating groups, unlike symmetric groups, contain only even permutations. The user types in the degree of the desired alternating group and clicks Generate Alternating Group. If the degree is d, the order of the resulting alternating group Ad is d!/2, and the resulting Cayley table has (d!/2)2 entries. Because the sizes of these groups and tables increase so quickly, GALOIS only accommodates degrees in the range 1-5.
If requested, the display of the generated group is stored in an Alternating Group View File in html format. If a file with the chosen filename already exists, the output will be appended following previous output; otherwise, a new file is created. This View is then displayed in a pop-up browser window.
The Alternating Group View may include any or all of the components shown below. (For more information, see the Symmetric Group Analyzer Panel.)
An Alternating Group View will also be generated if the alternating group is isomorphic to a solution analyzed on the Solution Analysis Panel, if the user clicks that View Group button.
The Permutation Group Analyzer Panel is accessed from the Special Groups menu on the Main Screen. Although its output resembles the Symmetric Group Analyzer Panel and the Alternating Group Analyzer Panel, this panel is more general, enabling the user to generate all kinds of permutation groups, provided that the group's order does not exceed 120 and its degree does not exceed 15. The user first enters the desired degree and clicks Continue. An array of slots appears, into which desired permutations can be entered directly. (The identity permutation, which is required to constitute a valid group, is already predefined on the screen and cannot be modified.) The number of slots depends on the degree, up to the 120-slot limit if the degree is greater than 4. In the latter case, the scrolling feature (common to all GALOIS screens) can be used to move left, right, up, and down through the slots.
Each permutation is entered as a sequence of digits; if the degree is greater than 9, however, positions 10-15 are represented by the letters A-F (which may be typed in either upper or lower case). For example, the permutation that reverses an entire sequence of 11 objects can be entered by typing BA987654321. After entering permutations, the user clicks the View Permutation Group button. If any duplicate permutations have been entered, a warning message will appear and they will be eliminated, and any other entry errors will be flagged, with an explanatory message provided. By default, GALOIS will generate any additional elements that are needed to complete the group, up to the limit of 120 elements. Therefore the user only needs to enter a set of generators for the desired group. Alternatively, the user can click an option that limits the elements to those that have been explicitly entered; under that option, an error message will be returned if any needed elements are missing from the group. If necessary, all entered permutations can be erased by clicking a Clear Elements button.
When a complete valid group has been defined, it is stored (if requested by the Create View checkbox) in a Permutation Group View File in html format. If a file with the chosen filename already exists, the output will be appended following previous output; otherwise, a new file is created. The Permutation Group View is displayed in a pop-up browser window.
The Permutation Group View may include any or all of the components shown below. (For more information, see the Symmetric Group Analyzer Panel.)
A Permutation Group View can also be generated from the Solution Analysis Panel. Every solution group is isomorphic to the permutation group defined by its rows or its columns, and a View of that group is created by clicking on the corresponding View Group button in the solution analysis.
The Additive Integer Modulo N Group Analyzer Panel is accessed from the Special Groups menu on the Main Screen and enables the user to generate additive integer groups modulo n. The user types in the modulus (1-120) for the desired additive group and clicks Generate Additive Group. The display of the generated group is stored (if Create View was requested) in an Additive Group View File in html format. If a file with the chosen filename already exists, the output will be appended following previous output; otherwise, a new file is created. This View is then displayed in a pop-up browser window.
The Additive Group View may include any or all of the components shown below:
An Additive Group View can also be generated from the Solution Analysis Panel if the solution is a cyclic group. Every cyclic group of order n is isomorphic to Zn, and a View of the latter is created by clicking on the corresponding View Group button in the solution analysis.
The Multiplicative Integer Modulo N Group Analyzer Panel is accessed from the Special Groups menu on the Main Screen and enables the user to generate multiplicative integer groups modulo n. This screen offers two options: The user can enter either a modulus (2-1000) or an order (1-120) for the desired multiplicative group(s), before clicking a Compute button. If a modulus was entered, GALOIS computes and displays the group's order (number of elements), and if that order does not exceed the program limit of 120 elements, a View Group button is displayed. If an order was entered, GALOIS displays a list of multiplicative integer groups of the desired order (up to a limit of 20 groups), with a View Group button for each.
The display of the generated group is stored (if Create View was requested) in a Multiplicative Group View File in html format. If a file with the chosen filename already exists, the output will be appended following previous output; otherwise, a new file is created. This View is then displayed in a pop-up browser window.
The Multiplicative Group View may include any or all of the components shown below:
An Multiplicative Group View can also be generated from the Solution Analysis Panel if the solution is isomorphic to one or more such groups. A view of the multiplicative group is created by clicking on the corresponding View Group button in the solution analysis.
The Direct Product Group Analyzer Panel is accessed from the Special Groups menu on the Main Screen and enables the user to generate direct product groups based on two to six component groups. Each component may be either an additive group, defined by entering a modulus (2-60), or any other group previously generated by GALOIS, indicated by clicking a Choose Group button and selecting a group from the displayed Generated Group List. Any combination of additive groups and previously generated groups can be specified, provided that the order of the resulting group (i. e., the product of the orders of its components) does not exceed the program limit of 120. A Clear Components button enables the user to restart following an entry error.
The display of the generated group is stored (if Create View was requested) in a Direct Product Group View File in html format. If a file with the chosen filename already exists, the output will be appended following previous output; otherwise, a new file is created. This View is then displayed in a pop-up browser window.
The Direct Product Group View may include any or all of the outputs shown below:
A Direct Product Group View may also be generated from the Solution Analysis Panel if the solution is isomorphic to one or more such groups. A view of the direct product group is created by clicking on the corresponding View Group button in the solution analysis.
The Automorphism Group Analyzer Panel is accessed from the Special Groups menu on the Main Screen and enables the user to generate the automorphism group (up to a limit of 120 automorphisms) and/or the inner automorphism group derived from any previously generated group. The generated group on which automorphisms are to be computed is specified by clicking on a Choose Group button and selecting a group from the displayed Generated Group List. GALOIS responds by listing the automorphisms, beginning with the inner automorphisms, up to the program limit of 120, where each automorphism is represented as a permutation and accompanied by a Show Automorphism button, which can be clicked to view that automorphism in more detail on the Automorphism Panel. The user can then create the desired group by clicking Generate Automorphism Group or Generate Inner Automorphism Group.
The display of the generated group is stored (if Create View was requested) in an Automorphism Group View File in html format. If a file with the chosen filename already exists, the output will be appended following previous output; otherwise, a new file is created. This View is then displayed in a pop-up browser window.
The Automorphism Group View may include any or all of the outputs shown below. (For more information, see the Symmetric Group Analyzer Panel.)
The Automorphism Panel is invoked when the user clicks the Show Automorphism button next to an automorphism listed on the Automorphism Group Analyzer Panel. An automorphism is a function f that establishes a one-to-one correspondence between the elements of a group and some permutation of those elements, such that the fundamental group structure is preserved by the function. Specifically, f(x) . f(y) must equal f(x . y) for any elements x, y in the group. This panel exhibits the function f by indicating the specific element in the group to which each element is mapped. The group's Cayley table is displayed so that the equality between each f(x . y) and the corresponding f(x) . f(y) can be readily verified.
The Cayley table can be displayed either in conventional Row and Column Format or in a more compact Calculator Format, and elements can be represented either by their original names or their reductive-mapping names. If Calculator Format is used, then the product of two elements can be determined by clicking on those elements on two drop-down lists. By default, large tables are displayed in Calculator Format and smaller ones in Row and Column Format, while the elements are represented by their original names. These defaults can be overridden by means of the Table Format menu (or permanently through the Preferences Panel).
The Preferences Panel is accessed from the menu on the Main Screen. It includes the following tabs:
The Analysis Tool Panels are reached through the Analysis Tools menu on the Main Screen. These panels enable the user to examine, analyze, and compare the various groups that have been generated either through a search from the Main Screen or through one of the Special Group Panels. If the Autosave option is used, these panels can access not only the groups created in the current GALOIS session, but also groups created in previous sessions. Groups are referenced by Group IDs, assigned by GALOIS when each group was generated. The various categories of Group IDs are listed in the Appendix.
The Generated Group Analysis Panels include the following:
Several of these panels may lead to long-running processes, particularly where groups of more than 60 elements are involved. Where appropriate, GALOIS provided a Halt button allowing the user to cancel such processes.
The Generated Group List Panel can be reached through the Analysis Tools menu on the Main Screen. These panel lists all of the groups that have been generated by GALOIS either through a search from the Main Screen or through one of the Special Group Panels. By default (unless overridden by the Preferences Panel), the groups are listed in the order they were generated, but an option is provided to re-sort the list in alphabetical order. Each group is identified by its Group ID, assigned by GALOIS when the group was generated. The various categories of Group IDs are listed in the Appendix. After clicking on a Group ID, the user can click Show Group to view the group on the Group Display Panel.
The Generated Group List Panel is also displayed whenever the user clicks on Choose Group from another panel in GALOIS. The user clicks on the desired Group ID and then on Choose Group to return to the previous panel.
The Group Display Panel is reached by clicking on a Group ID on the Generated Group List Panel and then clicking Show Group. The panel displays the order of the selected group, its Cayley table, and its reductive mapping. For relatively simple groups, including all cyclic and dihedral groups, the panel includes a Show Graph button, giving access to a Graphic Representation Panel showing the group structure in visual form.
The Cayley table can be viewed either in conventional Row and Column Format or in a more compact Calculator Format, and elements can be represented either by their original names or their reductive-mapping names. If Calculator Format is used, then the product of two elements can be determined by clicking on those elements on two drop-down lists. By default, large tables are displayed in Calculator Format and smaller ones in Row and Column Format, while the elements are represented by their original names. These defaults can be overridden by means of the Table Format menu (or permanently through the Preferences Panel).
After examining the group, the user clicks OK to return to the Generated Group List Panel.
The Group Comparison Panel, reached through the Analysis Tools menu on the Main Screen, examines any two previously generated groups for relationships based on isomorphism or homomorphism. The groups are specified by clicking on two Choose Group buttons, each opening a Generated Group List Panel from which that group is to be selected. GALOIS then compares the two groups as follows:
After examining these relationships, the user can use one of the Choose Group buttons to compare against a different group or click on Restart to clear both groups from the panel and begin again. When finished, the user clicks OK to return to the Main Screen.
*Additional homomorphisms can be determined by taking the composite function of this homomorphism and an automorphism on either the larger or the smaller group. For example, if a is an automorphism on group G and h is a homomorphism from group G to group H, then h o a, which maps an element x into h(a(x)), is also a homomorphism from G to H. Automorphisms can be located through the Automorphism Group Analyzer Panel.
The Homomorphism Panel is invoked when the user clicks the Show Homomorphism button next to a homomorphism identified by the Group Comparison Panel. A homomorphism is a function f that maps one group into another, such that the fundamental group structure is preserved by the function. Specifically, if the first group's operation is labeled . and the second group's operation is ~, then f(x)~f(y) must equal f(x . y) for any x, y in the first group. (Note that a homomorphism, unlike an isomorphism, is not necessarily a one-to-one correspondence.) This panel identifies such a function f by indicating the specific element in the second group to which each element in the first group is mapped. The two Cayley tables are displayed side by side so that the equality between each f(x . y) and the corresponding f(x)~f(y) can be readily verified.
The Cayley tables on this screen can be displayed either in conventional Row and Column Format or in a more compact Calculator Format, and elements can be represented either by their original names or their reductive-mapping names. If Calculator Format is used, then the product of two elements can be determined by clicking on those elements on two drop-down lists. By default, large tables are displayed in Calculator Format and smaller ones in Row and Column Format, while the elements are represented by their original names. These defaults can be overridden by means of the Table Format menu (or permanently through the Preferences Panel).
The Conjugacy Class Panel, reached through the Analysis Tools menu on the Main Screen, lists the conjugacy classes for a given group and their component elements and computes the terms in the group's class equation. The group is specified by clicking on a Choose Group button, which opens a Generated Group List Panel from which the group is to be selected. The conjugacy classes are identified one by one and their elements are listed. Each conjugacy class Cl(g) consists of all those elements that are conjugate to a particular element g. Since an element is conjugate to g if and only if it can be written as x-1gx for some other element x, the element is also expressed in this form.
If the group is a permutation group, then the element's cycle decomposition is also shown, along with the partition of the group's degree defined by that decomposition. Within a symmetric group, permutations belong to the same conjugacy class if and only if they partition the degree in the same manner. (Within a subgroup, such as an alternating group, permutations in the same conjugacy class are always associated with the same partition, but not conversely.) The example below shows two permutations of degree 5. The first permutation can be written as the product of the cycles (1), (23), and (45). (The cycle (1), which is equivalent to the identity element and does not affect the product, is omitted by GALOIS.) This decomposition thus partitions degree 5 into 1 + 2 + 2. The second permutation produces the same 1 + 2 + 2 partition (following the convention that partitions are written in ascending order), so the two permutations belong to the same conjugacy class of S5.
| Example: | Permutation | Decomposition into Cycles | Partition | ||
| ( | 12345 13254 | ) | (23) (45) | 5 = 1 + 2 + 2 | |
| ( | 12345 21435 | ) | (12) (34) | 5 = 1 + 2 + 2 | |
The class equation is worked out at the bottom of the screen. Because the class equation uses the orders of centralizer subgroups, the user may also wish to refer to the Subgroup Panel, which includes a calculator for these subgroups. After examining the conjugacy classes and class equation, the user can click Choose Group again to call up the same information for a different group or OK to return to the Main Screen.
From time to time, it may become desirable to remove groups that are no longer needed from the Objects File, where groups generated by GALOIS are stored. The Generated Groups menu on the Main Screen offers options to delete selected groups or to delete all groups from the file and from program memory. If the "Delete All Groups..." option is selected, the Objects File is copied to the Backup Objects File, after which all generated groups are removed from the Objects File. The "Delete Selected Groups..." option works in the same manner, except that only the groups selected on a Group Deletion Panel are removed.
The Group Deletion Panel presents a "Retain List" and a "Delete List" side by side. Initially, the Delete List is empty, while the Retain List lists all of the groups that have been generated by GALOIS. By default (unless overridden by the Preferences Panel), the Retain List shows groups in the order they were generated, but an option is provided to re-sort the list in alphabetical order. Each group is identified by its Group ID, assigned by GALOIS when the group was generated. The user clicks on a Group ID to move it from either panel to the other. If a group is moved to the Delete List, then any other groups that are dependent on it are also automatically moved (after a warning) to the Delete List, where they are flagged by asterisks. For example, if Group#6-1 is to be deleted and Subgroup3-1(Group#6-1) is in the file, the subgroup is moved to the Delete List along with its parent group. (For an exposition of the various categories of Group IDs, see the Appendix.) Similarly, the subgroup cannot be returned to the Retain List until its parent group has been returned there. After building and reviewing the Delete List, the user clicks on Delete Selected Groups, and the groups are removed both from memory and from the permanent Objects File. The user can exit from the panel at any point by clicking on Exit.
GALOIS uses external files to record user preferences and to retain generated groups from session to session. In addition, various panels within GALOIS produce View Files in a convenient html format, enabling the user to review generated groups even after exiting from the application.
The Autosave option allows the user to retain groups that are generated in one GALOIS session for use in future sessions. If this option is in effect when a GALOIS session is started, the program reads an Objects File (GaloisObjects.bas, residing in the same directory as the GALOIS executable) and rebuilds any groups that may have previously been recorded; as additional groups are generated, they are added to this file. As the file grows in size, start-up time for the application may increase. This problem can be remedied by deleting some or all of the generated groups (see the Group Deletion Panel). The Autosave option can also be turned off entirely using the Preferences Panel.
Before deleting groups, GALOIS copies the Objects File to an Objects Backup File (GaloisObjectsBackup.bas). If an error is encountered while reading groups from the Objects File at start-up, GALOIS also copies the file to this backup and then attempts to salvage groups to the extent possible from the Objects File. In the event of major user error or system error, the groups in the Backup File may be recovered by removing GaloisObjects.bas and copying the Objects Backup file into a new GaloisObjects.bas file.
The Preferences File (GaloisPreferences.bas, residing in the same directory as the GALOIS executable) is created if and when the user first enters preferences into the Preferences Panel. If no Preferences File exists in the directory when GALOIS is opened, the system defaults that are coded into the program will be used.
View Files are created by various panels throughout GALOIS, as requested by the user. These files exhibit information about generated groups in a convenient html format and can be reviewed even after the user has exited from the application. The default name or directory for each View File can be overridden by the user through a "Save as" dialog box when the file is created.
where the ai are representative elements from the n conjugacy classes of G, and Z(ai) is the centralizer of ai. The equation derives from the fact that every element in a group must belong to one and only one conjugacy class. The elements in the conjugacy class containing ai are in one-to-one correspondence with the right cosets of Z(ai); hence the size of the whole class is given by o(G)/o(Z(ai)).
| ( | 12345 14523 | ) | = (24) (35) |
The various groups generated by searching from the Main Screen, by the analysis panels, and by the panels listed on the Special Groups menu are retained and assigned unique Group IDs by which they can be referenced in other program functions, including in particular the functions listed on the Analysis Tools menu. For example, a list of all the generated groups generated is provided on the Generated Group List Panel; a display of an individual group can then be obtained by clicking on the Show Group button next to its Group ID. If the Autosave option is used, the Group IDs uniquely identify not only the groups created in the current GALOIS session, but also groups created in previous sessions.
The various categories of identifiers are indicated below.
| GROUP ID | DESCRIPTION |
| An | The alternating group of degree n. |
| Aut(sourcegroup) | The automorphism group whose elements are the automorphisms of sourcegroup. |
| Columns(sourcegroup) | The permutation group defined by the columns in sourcegroup's Cayley table. (See the Solution Analysis Panel.) |
| Group #n-m | The mth solution group of group order n created through a search from the Main Screen where the search did not include isomorphic equivalents. |
| Group #n(m) | The mth solution group of group order n created through a search from the Main Screen where the Include Isomorphic Equivalents checkbox was checked prior to the search. |
| group-1 x group-2 | The direct product of group-1 and group-2. |
| group-1 x group-2 ... x group-n | The direct product of the three or more groups group-1 through group-n. |
| Inn(sourcegroup) | The inner automorphism group whose elements are the inner automorphisms of sourcegroup. |
| Manualn-m | The mth solution group created by manual entry into the textboxes on the Main Screen. The group order is given by n. |
| Nn(parentgroup) | The nth subgroup in parentgroup's composition series. N1(parentgroup) is the maximal normal subgroup of parentgroup, and each succeeding member of the series is the maximal normal subgroup of its predecessor. |
| Permn-m | The mth permutation group created by the Permutation Group Analyzer Panel. The group order is given by n. |
| parentgroup/Subgroupn-m | The quotient group consisting of Subgroupn-m(parentgroup) and its cosets within parentgroup. |
| Rows(sourcegroup) | The permutation group defined by the rows in sourcegroup's Cayley table. (See the Solution Analysis Panel.) |
| Sn | The symmetric group of degree n. |
| Subgroupn-m(parentgroup) | The mth subgroup of order n within group parentgroup. |
| Zn | The additive group of integers modulo n. |
| Zn* | The multiplicative group of integers modulo n. |