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### 2 A First Session with GAP

This tutorial introduces you to the GAP system. It is written with users in mind who have just managed to start GAP for the first time on their computer and want to learn the basic facts about GAP by playing around with some instructive examples. Therefore, this tutorial contains at many places examples consisting of several lines of input (which you should type on your terminal) followed by the corresponding output ( which GAP produces as an answer to your input).

We encourage you to actually run through these examples on your computer. This will support your feeling for GAP as a tool, which is the leading aim of this tutorial. Do not believe any statement in it as long as you cannot verify it for your own version of GAP. You will learn to distinguish between small deviations of the behavior of your personal GAP from the printed examples and serious nonsense.

Since the printing routines of GAP are in some sense machine dependent you will for instance encounter a different layout of the printed objects in different environments. But the contents should always be the same. In case you encounter serious nonsense it is highly recommended that you send a bug report to support@gap-system.org.

The examples in this tutorial should explain everything you have to know in order to be able to use GAP. The reference manual then gives a more systematic treatment of the various types of objects that GAP can manipulate. It seems desirable neither to start this systematic course with the most elementary (and most boring) structures, nor to confront you with all the complex data types before you know how they are composed from elementary structures. For this reason this tutorial wants to provide you with a basic understanding of GAP objects, on which the reference manual will then build when it explains everything in detail. So after having mastered this tutorial, you can immediately plunge into the exciting parts of GAP and only read detailed information about elementary things (in the reference manual) when you really need them.

Each chapter of this tutorial contains a section with references to the reference manual at the end.

#### 2.1 Starting and Leaving GAP

If the program is correctly installed then you usually start GAP by simply typing gap at the prompt of your operating system followed by the key, sometimes this is also called the key.

$gap  GAP answers your request with its beautiful banner and then it shows its own prompt gap> asking you for further input. (You can avoid the banner with the command line option -b; more command line options are described in Section Reference: Command Line Options.) gap>  The usual way to end a GAP session is to type quit; at the gap> prompt. Do not omit the semicolon! gap> quit;$


On some systems you could type to yield the same effect. In any situation GAP is ended by typing twice within a second. Here as always, a combination like means that you have to press the key while you hold down the key.

On some systems minor changes might be necessary. This is explained in GAP installation instructions (see the INSTALL file in the GAP root directory, or the GAP website).

In most places whitespace characters (i.e. s, s and s) are insignificant for the meaning of GAP input. Identifiers and keywords must however not contain any whitespace. On the other hand, sometimes there must be whitespace around identifiers and keywords to separate them from each other and from numbers. We will use whitespace to format more complicated commands for better readability.

A comment in GAP starts with the symbol # and continues to the end of the line. Comments are treated like whitespace by GAP. We use comments in the printed examples in this tutorial to explain certain lines of input or output.

#### 2.2 Loading Source Code from a File

The most convenient way of creating larger pieces of GAP code is to write them to some text file. For this purpose you can simply use your favorite text editor. You can load such a file into GAP using the Read (Reference: Read) function:

gap> Read("../../GAPProgs/Example.g");


You can either give the full absolute path name of the source file or its relative path name from the current directory.

#### 2.3 The Read Evaluate Print Loop

GAP is an interactive system. It continuously executes a read evaluate print loop. Each expression you type at the keyboard is read by GAP, evaluated, and then the result is shown.

The interactive nature of GAP allows you to type an expression at the keyboard and see its value immediately. You can define a function and apply it to arguments to see how it works. You may even write whole programs containing lots of functions and test them without leaving the program.

When your program is large it will be more convenient to write it on a file and then read that file into GAP. Preparing your functions in a file has several advantages. You can compose your functions more carefully in a file (with your favorite text editor), you can correct errors without retyping the whole function and you can keep a copy for later use. Moreover you can write lots of comments into the program text, which are ignored by GAP, but are very useful for human readers of your program text. GAP treats input from a file in the same way that it treats input from the keyboard. Further details can be found in section Read (Reference: Read).

A simple calculation with GAP is as easy as one can imagine. You type the problem just after the prompt, terminate it with a semicolon and then pass the problem to the program with the key. For example, to multiply the difference between 9 and 7 by the sum of 5 and 6, that is to calculate (9 - 7) * (5 + 6), you type exactly this last sequence of symbols followed by ; and .

gap> (9 - 7) * (5 + 6);
22
gap>


Then GAP echoes the result 22 on the next line and shows with the prompt that it is ready for the next problem. Henceforth, we will no longer print this additional prompt.

If you make a mistake while typing the line, but before typing the final , you can use the key (or sometimes key) to delete the last typed character. You can also move the cursor back and forward in the line with and and insert or delete characters anywhere in the line. The line editing commands are fully described in section Reference: Line Editing.

If you did omit the semicolon at the end of the line but have already typed , then GAP has read everything you typed, but does not know that the command is complete. The program is waiting for further input and indicates this with a partial prompt >. This problem is solved by simply typing the missing semicolon on the next line of input. Then the result is printed and the normal prompt returns.

gap> (9 - 7) * (5 + 6)
> ;
22


So the input can consist of several lines, and GAP prints a partial prompt > in each input line except the first, until the command is completed with a semicolon. (GAP may already evaluate part of the input when is typed, so for long calculations it might take some time until the partial prompt appears.) Whenever you see the partial prompt and you cannot decide what GAP is still waiting for, then you have to type semicolons until the normal prompt returns. In every situation the exact meaning of the prompt gap> is that the program is waiting for a new problem.

But even if you mistyped the command more seriously, you do not have to type it all again. Suppose you mistyped or forgot the last closing parenthesis. Then your command is syntactically incorrect and GAP will notice it, incapable of computing the desired result.

gap> (9 - 7) * (5 + 6;
Syntax error: ) expected
(9 - 7) * (5 + 6;
^


Instead of the result an error message occurs indicating the place where an unexpected symbol occurred with an arrow sign ^ under it. As a computer program cannot know what your intentions really were, this is only a hint. But in this case GAP is right by claiming that there should be a closing parenthesis before the semicolon. Now you can type to recover the last line of input. It will be written after the prompt with the cursor in the first position. Type to take the cursor to the end of the line, then to move the cursor one character back. The cursor is now on the position of the semicolon. Enter the missing parenthesis by simply typing ). Now the line is correct and may be passed to GAP by hitting the key. Note that for this action it is not necessary to move the cursor past the last character of the input line.

Each line of commands you type is sent to GAP for evaluation by pressing regardless of the position of the cursor in that line. We will no longer mention the key from now on.

Sometimes a syntax error will cause GAP to enter a break loop. This is indicated by the special prompt brk>. If another syntax error occurs while GAP is in a break loop, the prompt will change to brk_02>, brk_03> and so on. You can leave the current break loop and exit to the next outer one by either typing quit; or by hitting . Eventually GAP will return to its normal state and show its normal prompt gap> again.

#### 2.4 Constants and Operators

In an expression like (9 - 7) * (5 + 6) the constants 5, 6, 7, and 9 are being composed by the operators +, * and - to result in a new value.

There are three kinds of operators in GAP, arithmetical operators, comparison operators, and logical operators. You have already seen that it is possible to form the sum, the difference, and the product of two integer values. There are some more operators applicable to integers in GAP. Of course integers may be divided by each other, possibly resulting in noninteger rational values.

gap> 12345/25;
2469/5


Note that the numerator and denominator are divided by their greatest common divisor and that the result is uniquely represented as a division instruction.

The next self-explanatory example demonstrates negative numbers.

gap> -3; 17 - 23;
-3
-6


The exponentiation operator is written as ^. This operation in particular might lead to very large numbers. This is no problem for GAP as it can handle numbers of (almost) any size.

gap> 3^132;
955004950796825236893190701774414011919935138974343129836853841


The mod operator allows you to compute one value modulo another.

gap> 17 mod 3;
2


Note that there must be whitespace around the keyword mod in this example since 17mod3 or 17mod would be interpreted as identifiers. The whitespace around operators that do not consist of letters, e.g., the operators * and -, is not necessary.

GAP knows a precedence between operators that may be overridden by parentheses.

gap> (9 - 7) * 5 = 9 - 7  * 5;
false


Besides these arithmetical operators there are comparison operators in GAP. A comparison results in a boolean value which is another kind of constant. The comparison operators =, <>, <, <=, > and >=, test for equality, inequality, less than, less than or equal, greater than and greater than or equal, respectively.

gap> 10^5 < 10^4;
false


The boolean values true and false can be manipulated via logical operators, i. e., the unary operator not and the binary operators and and or. Of course boolean values can be compared, too.

gap> not true; true and false; true or false;
false
false
true
gap> 10 > 0 and 10 < 100;
true


Another important type of constants in GAP are permutations. They are written in cycle notation and they can be multiplied.

gap> (1,2,3);
(1,2,3)
gap> (1,2,3) * (1,2);
(2,3)


The inverse of the permutation (1,2,3) is denoted by (1,2,3)^-1. Moreover the caret operator ^ is used to determine the image of a point under a permutation and to conjugate one permutation by another.

gap> (1,2,3)^-1;
(1,3,2)
gap> 2^(1,2,3);
3
gap> (1,2,3)^(1,2);
(1,3,2)


The various other constants that GAP can deal with will be introduced when they are used, for example there are elements of finite fields such as Z(8), and complex roots of unity such as E(4).

The last type of constants we want to mention here are the characters, which are simply objects in GAP that represent arbitrary characters from the character set of the operating system. Character literals can be entered in GAP by enclosing the character in singlequotes '.

gap> 'a';
'a'
gap> '*';
'*'


There are no operators defined for characters except that characters can be compared.

In this section you have seen that values may be preceded by unary operators and combined by binary operators placed between the operands. There are rules for precedence which may be overridden by parentheses. A comparison results in a boolean value. Boolean values are combined via logical operators. Moreover you have seen that GAP handles numbers of arbitrary size. Numbers and boolean values are constants. There are other types of constants in GAP like permutations. You are now in a position to use GAP as a simple desktop calculator.

#### 2.5 Variables versus Objects

The constants described in the last section are specified by certain combinations of digits and minus signs (in the case of integers) or digits, commas and parentheses (in the case of permutations). These sequences of characters always have the same meaning to GAP. On the other hand, there are variables, specified by a sequence of letters and digits (including at least one letter), and their meaning depends on what has been assigned to them. An assignment is done by a GAP command sequence_of_letters_and_digits := meaning, where the sequence on the left hand side is called the identifier of the variable and it serves as its name. The meaning on the right hand side can be a constant like an integer or a permutation, but it can also be almost any other GAP object. From now on, we will use the term object to denote something that can be assigned to a variable.

There must be no whitespace between the : and the = in the assignment operator. Also do not confuse the assignment operator with the single equality sign = which in GAP is only used for the test of equality.

gap> a:= (9 - 7) * (5 + 6);
22
gap> a;
22
gap> a * (a + 1);
506
gap> a = 10;
false
gap> a:= 10;
10
gap> a * (a + 1);
110


After an assignment the assigned object is echoed on the next line. The printing of the object of a statement may be in every case prevented by typing a double semicolon.

gap> w:= 2;;


After the assignment the variable evaluates to that object if evaluated. Thus it is possible to refer to that object by the name of the variable in any situation.

This is in fact the whole secret of an assignment. An identifier is bound to an object and from this moment points to that object. Nothing more. This binding is changed by the next assignment to that identifier. An identifier does not denote a block of memory as in some other programming languages. It simply points to an object, which has been given its place in memory by the GAP storage manager. This place may change during a GAP session, but that doesn't bother the identifier. The identifier points to the object, not to a place in the memory.

For the same reason it is not the identifier that has a type but the object. This means on the other hand that the identifier a which now is bound to an integer object may in the same session point to any other object regardless of its type.

Identifiers may be sequences of letters and digits containing at least one letter. For example abc and a0bc1 are valid identifiers. But also 123a is a valid identifier as it cannot be confused with any number. Just 1234 indicates the number 1234 and cannot be at the same time the name of a variable.

Since GAP distinguishes upper and lower case, a1 and A1 are different identifiers. Keywords such as quit must not be used as identifiers. You will see more keywords in the following sections.

In the remaining part of this manual we will ignore the difference between variables, their names (identifiers), and the objects they point to. It may be useful to think from time to time about what is really meant by terms such as "the integer w".

There are some predefined variables coming with GAP. Many of them you will find in the remaining chapters of this manual, since functions are also referred to via identifiers.

You can get an overview of all GAP variables by entering NamesGVars(). Many of these are predefined. If you are interested in the variables you have defined yourself in the current GAP session, you can enter NamesUserGVars().

gap> NamesUserGVars();
[ "a", "w" ]


This seems to be the right place to state the following rule: The name of every global variable in the GAP library starts with a capital letter. Thus if you choose only names starting with a small letter for your own variables you will not attempt to overwrite any predefined variable. (Note that most of the predefined variables are read-only, and trying to change their values will result in an error message.)

There are some further interesting variables one of which will be introduced now.

Whenever GAP returns an object by printing it on the next line this object is assigned to the variable last. So if you computed

gap> (9 - 7) * (5 + 6);
22


and forgot to assign the object to the variable a for further use, you can still do it by the following assignment.

gap> a:= last;
22


Moreover there are variables last2 and last3, you can guess their values.

In this section you have seen how to assign objects to variables. These objects can later be accessed through the name of the variable, its identifier. You have also encountered the useful concept of the last variables storing the latest returned objects. And you have learned that a double semicolon prevents the result of a statement from being printed.

#### 2.6 Objects vs. Elements

In the last section we mentioned that every object is given a certain place in memory by the GAP storage manager (although that place may change in the course of a GAP session). In this sense, objects at different places in memory are never equal, and if the object pointed to by the variable a (to be more precise, the variable with identifier a) is equal to the object pointed to by the variable b, then we should better say that they are not only equal but identical. GAP provides the function IsIdenticalObj (Reference: IsIdenticalObj) to test whether this is the case.

gap> a:= (1,2);; IsIdenticalObj( a, a );
true
gap> b:= (1,2);; IsIdenticalObj( a, b );
false
gap> b:= a;; IsIdenticalObj( a, b );
true


As the above example indicates, GAP objects a and b can be unequal although they are equal from a mathematical point of view, i.e., although we should have a = b. It may be that the objects a and b are stored in different places in memory, or it may be that we have an equivalence relation defined on the set of objects under which a and b belong to the same equivalence class. For example, if a = x^3 and b = x^{-5} are words in the finitely presented group ⟨ x ∣ x^2 = 1 ⟩, we would have a = b in that group.

GAP uses the equality operator = to denote such a mathematical equality, not the identity of objects. Hence we often have a = b although IsIdenticalObj( a, b ) = false. The operator = defines an equivalence relation on the set of all GAP objects, and we call the corresponding equivalence classes elements. Phrasing it differently, the same element may be represented by various GAP objects.

Non-trivial examples of elements that are represented by different objects (objects that really look different, not ones that are merely stored in different memory places) will occur only when we will be considering composite objects such as lists or domains.

#### 2.7 About Functions

A program written in the GAP language is called a function. Functions are special GAP objects. Most of them behave like mathematical functions. They are applied to objects and will return a new object depending on the input. The function Factorial (Reference: Factorial), for example, can be applied to an integer and will return the factorial of this integer.

gap> Factorial(17);
355687428096000


Applying a function to arguments means to write the arguments in parentheses following the function. Several arguments are separated by commas, as for the function Gcd (Reference: Gcd) which computes the greatest common divisor of two integers.

gap> Gcd(1234, 5678);
2


There are other functions that do not return an object but only produce a side effect, for example changing one of their arguments. These functions are sometimes called procedures. The function Print (Reference: Print) is only called for the side effect of printing something on the screen.

gap> Print(1234, "\n");
1234


In order to be able to compose arbitrary text with Print (Reference: Print), this function itself will not produce a line break after printing. Thus we had another newline character "\n" printed to start a new line.

Some functions will both change an argument and return an object such as the function Sortex (Reference: Sortex) that sorts a list and returns the permutation of the list elements that it has performed. You will not understand right now what it means to change an object. We will return to this subject several times in the next sections.

A comfortable way to define a function yourself is the maps-to operator -> consisting of a minus sign and a greater sign with no whitespace between them. The function cubed which maps a number to its cube is defined on the following line.

gap> cubed:= x -> x^3;
function( x ) ... end


After the function has been defined, it can now be applied.

gap> cubed(5);
125


More complicated functions, especially functions with more than one argument cannot be defined in this way. You will see how to write your own GAP functions in Section 4.1.

In this section you have seen GAP objects of type function. You have learned how to apply a function to arguments. This yields as result a new object or a side effect. A side effect may change an argument of the function. Moreover you have seen an easy way to define a function in GAP with the maps-to operator.

#### 2.8 Help

The content of the GAP manuals is also available as on-line help. A GAP session loads a long list of index entries. This typically contains all chapter and section headers, all names of documented functions, operations and so on, as well as some explicit index entries defined in the manuals.

The format of a query is as follows.

?[book:][?]topic

A simple example would be to type ?help at the GAP prompt. If there is a single section with index entry topic then this is displayed directly.

If there are several matches you get an overview like in the example below.

gap> ?sets
Help: several entries match this topic - type ?2 to get match [2]

[1] Tutorial: Sets
[2] Reference: Sets
[3] Reference: sets
[4] Reference: Sets of Subgroups
[5] Reference: setstabilizer


GAP's manuals consist of several books, which are indicated before the colon in the list above. A help query can be restricted to one book by using the optional book: part. For example ?tut : sets will display the first of these help sections. More precisely, the parts of the string book which are separated by white space are interpreted as beginnings of the first words in the name of the book. Try ?books to see the list of available books and their names.

The search for a matching topic (and optional book) is done case insensitively. If there is another ? before the topic, then a substring search for topic is performed on all index entries. Otherwise the parts of topic which are separated by white space are considered as beginnings of the first words in an index entry.

White space is normalized in the search string (and the index entries).

Examples. All the following queries lead to the chapter of the reference manual which explains the use of GAP's help system in more detail.

gap> ?Reference: The Help System
gap> ?  REF : t h s
gap> ?ref:?  help   system


The query ??sets shows all help sections in all books whose index entries contain the substring sets.

As mentioned in the example above a complete list of commands for the help system is available in Section ?Ref: The Help System of the reference manual. In particular there are commands to browse through the help sections, see ?Ref: Browsing through the Sections and there is a way to influence the way how the help sections are displayed, see ?Ref: SetHelpViewer. For example you can use an external pager program, a Web browser, dvi-previewer and/or pdf-viewer for reading GAP's online help.

#### 2.9 Further Information introducing the System

For large amounts of input data, it might be advisable to write your input first into a file, and then read this into GAP; see Read (Reference: Read), Edit (Reference: Edit) for this.

The definition of the GAP syntax can be looked up in Chapter Reference: The Programming Language. A complete list of command line editing facilities is found in Section Reference: Line Editing. The break loop is described in Section Reference: Break Loops.

Operators are explained in more detail in Sections Reference: Expressions and Reference: Comparisons. You will find more information about boolean values in Chapters Reference: Booleans and Reference: Boolean Lists. Permutations are described in Chapter Reference: Permutations and characters in Chapter Reference: Strings and Characters.

Variables and assignments are described in more detail in Reference: Variables and Reference: Assignments. A complete list of keywords is contained in Reference: Keywords.

More about functions can be found in Reference: Function Calls and Reference: Procedure Calls.

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