Modern mathematics, especially algebra, is based on set theory. When sets are represented in a computer, they inadvertently turn into lists. That's why we start our survey of the various objects GAP can handle with a description of lists and their manipulation. GAP regards sets as a special kind of lists, namely as lists without holes or duplicates whose entries are ordered with respect to the precedence relation <.
After the introduction of the basic manipulations with lists in 3.1, some difficulties concerning identity and mutability of lists are discussed in 3.2 and 3.3. Sets, ranges, row vectors, and matrices are introduced as special kinds of lists in 3.4, 3.5, 3.8. Handy list operations are shown in 3.7. Finally we explain how to use records in 3.9.
A list is a collection of objects separated by commas and enclosed in brackets. Let us for example construct the list primes of the first ten prime numbers.
gap> primes:= [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]; [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ]
The next two primes are 31 and 37. They may be appended to the existing list by the function Append which takes the existing list as its first and another list as a second argument. The second argument is appended to the list primes and no value is returned. Note that by appending another list the object primes is changed.
gap> Append(primes, [31, 37]); gap> primes; [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 ]
You can as well add single new elements to existing lists by the function Add which takes the existing list as its first argument and a new element as its second argument. The new element is added to the list primes and again no value is returned but the list primes is changed.
gap> Add(primes, 41); gap> primes; [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 ]
Single elements of a list are referred to by their position in the list. To get the value of the seventh prime, that is the seventh entry in our list primes, you simply type
gap> primes[7]; 17
This value can be handled like any other value, for example multiplied by 2 or assigned to a variable. On the other hand this mechanism allows one to assign a value to a position in a list. So the next prime 43 may be inserted in the list directly after the last occupied position of primes. This last occupied position is returned by the function Length.
gap> Length(primes); 13 gap> primes[14]:= 43; 43 gap> primes; [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 ]
Note that this operation again has changed the object primes. The next position after the end of a list is not the only position capable of taking a new value. If you know that 71 is the 20th prime, you can enter it right now in the 20th position of primes. This will result in a list with holes which is however still a list and now has length 20.
gap> primes[20]:= 71; 71 gap> primes; [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,,,,,, 71 ] gap> Length(primes); 20
The list itself however must exist before a value can be assigned to a position of the list. This list may be the empty list [ ].
gap> lll[1]:= 2; Error, Variable: 'lll' must have a value gap> lll:= []; lll[1]:= 2; [ ] 2
Of course existing entries of a list can be changed by this mechanism, too. We will not do it here because primes then may no longer be a list of primes. Try for yourself to change the 17 in the list into a 9.
To get the position of 17 in the list primes use the function Position (Reference: Position) which takes the list as its first argument and the element as its second argument and returns the position of the first occurrence of the element 17 in the list primes. If the element is not contained in the list then Position (Reference: Position) will return the special object fail.
gap> Position(primes, 17); 7 gap> Position(primes, 20); fail
In all of the above changes to the list primes, the list has been automatically resized. There is no need for you to tell GAP how big you want a list to be. This is all done dynamically.
It is not necessary for the objects collected in a list to be of the same type.
gap> lll:= [true, "This is a String",,, 3]; [ true, "This is a String",,, 3 ]
In the same way a list may be part of another list.
gap> lll[3]:= [4,5,6];; lll; [ true, "This is a String", [ 4, 5, 6 ],, 3 ]
A list may even be part of itself.
gap> lll[4]:= lll; [ true, "This is a String", [ 4, 5, 6 ], ~, 3 ]
Now the tilde in the fourth position of lll denotes the object that is currently printed. Note that the result of the last operation is the actual value of the object lll on the right hand side of the assignment. In fact it is identical to the value of the whole list lll on the left hand side of the assignment.
A string is a special type of list, namely a dense list of characters, where dense means that the list has no holes. Here, characters are special GAP objects representing an element of the character set of the operating system. The input of printable characters is by enclosing them in single quotes '. A string literal can either be entered as the list of characters or by writing the characters between doublequotes ". Strings are handled specially by Print (Reference: Print). You can learn much more about strings in the reference manual.
gap> s1 := ['H','a','l','l','o',' ','w','o','r','l','d','.']; "Hallo world." gap> s1 = "Hallo world."; true gap> s1[7]; 'w'
Sublists of lists can easily be extracted and assigned using the operator list{ positions }.
gap> sl := lll{ [ 1, 2, 3 ] }; [ true, "This is a String", [ 4, 5, 6 ] ] gap> sl{ [ 2, 3 ] } := [ "New String", false ]; [ "New String", false ] gap> sl; [ true, "New String", false ]
This way you get a new list whose \(i\)-th entry is that element of the original list whose position is the \(i\)-th entry of the argument in the curly braces.
This second section about lists is dedicated to the subtle difference between equality and identity of lists. It is really important to understand this difference in order to understand how complex data structures are realized in GAP. This section applies to all GAP objects that have subobjects, e.g., to lists and to records. After reading the section 3.9 about records you should return to this section and translate it into the record context.
Two lists are equal if all their entries are equal. This means that the equality operator = returns true for the comparison of two lists if and only if these two lists are of the same length and for each position the values in the respective lists are equal.
gap> numbers := primes;; numbers = primes; true
We assigned the list primes to the variable numbers and, of course they are equal as they have both the same length and the same entries. Now we will change the third number to 4 and compare the result again with primes.
gap> numbers[3]:= 4;; numbers = primes; true
You see that numbers and primes are still equal, check this by printing the value of primes. The list primes is no longer a list of primes! What has happened? The truth is that the lists primes and numbers are not only equal but they are also identical. primes and numbers are two variables pointing to the same list. If you change the value of the subobject numbers[3] of numbers this will also change primes. Variables do not point to a certain block of storage memory but they do point to an object that occupies storage memory. So the assignment numbers := primes did not create a new list in a different place of memory but only created the new name numbers for the same old list of primes.
From this we see that the same object can have several names.
If you want to change a list with the contents of primes independently from primes you will have to make a copy of primes by the function ShallowCopy which takes an object as its argument and returns a copy of the argument. (We will first restore the old value of primes.)
gap> primes[3]:= 5;; primes; [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,,,,,, 71 ] gap> numbers:= ShallowCopy(primes);; numbers = primes; true gap> numbers[3]:= 4;; numbers = primes; false
Now numbers is no longer equal to primes and primes still is a list of primes. Check this by printing the values of numbers and primes.
Lists and records can be changed this way because GAP objects of these types have subobjects. To clarify this statement consider the following assignments.
gap> i:= 1;; j:= i;; i:= i+1;;
By adding 1 to i the value of i has changed. What happens to j? After the second statement j points to the same object as i, namely to the integer 1. The addition does not change the object 1 but creates a new object according to the instruction i+1. It is actually the assignment that changes the value of i. Therefore j still points to the object 1. Integers (like permutations and booleans) have no subobjects. Objects of these types cannot be changed but can only be replaced by other objects. And a replacement does not change the values of other variables. In the above example an assignment of a new value to the variable numbers would also not change the value of primes.
Finally try the following examples and explain the results.
gap> l:= [];; l:= [l]; [ [ ] ] gap> l[1]:= l; [ ~ ]
Now return to Section 3.1 and find out whether the functions Add (Reference: Add) and Append (Reference: Append) change their arguments.
GAP has a mechanism that protects lists against changes like the ones that have bothered us in Section 3.2. The function Immutable (Reference: Immutable) takes as argument a list and returns an immutable copy of it, i.e., a list which looks exactly like the old one, but has two extra properties: (1) The new list is immutable, i.e., the list itself and its subobjects cannot be changed. (2) In constructing the copy, every part of the list that can be changed has been copied, so that changes to the old list will not affect the new one. In other words, the new list has no mutable subobjects in common with the old list.
gap> list := [ 1, 2, "three", [ 4 ] ];; copy := Immutable( list );; gap> list[3][5] := 'w';; list; copy; [ 1, 2, "threw", [ 4 ] ] [ 1, 2, "three", [ 4 ] ] gap> copy[3][5] := 'w'; Lists Assignment: <list> must be a mutable list not in any function Entering break read-eval-print loop ... you can 'quit;' to quit to outer loop, or you can 'return;' and ignore the assignment to continue brk> quit;
As a consequence of these rules, in the immutable copy of a list which contains an already immutable list as subobject, this immutable subobject need not be copied, because it is unchangeable. Immutable lists are useful in many complex GAP objects, for example as generator lists of groups. By making them immutable, GAP ensures that no generators can be added to the list, removed or exchanged. Such changes would of course lead to serious inconsistencies with other knowledge that may already have been calculated for the group.
A converse function to Immutable (Reference: Immutable) is ShallowCopy (Reference: ShallowCopy), which produces a new mutable list whose \(i\)-th entry is the \(i\)-th entry of the old list. The single entries are not copied, they are just placed in the new list. If the old list is immutable, and hence the list entries are immutable themselves, the result of ShallowCopy (Reference: ShallowCopy) is mutable only on the top level.
It should be noted that also other objects than lists can appear in mutable or immutable form. Records (see Section 3.9) provide another example.
GAP knows several special kinds of lists. A set in GAP is a list that contains no holes (such a list is called dense) and whose elements are strictly sorted w.r.t. <; in particular, a set cannot contain duplicates. (More precisely, the elements of a set in GAP are required to lie in the same family, but roughly this means that they can be compared using the < operator.)
This provides a natural model for mathematical sets whose elements are given by an explicit enumeration.
GAP also calls a set a strictly sorted list, and the function IsSSortedList (Reference: IsSSortedList) tests whether a given list is a set. It returns a boolean value. For almost any list whose elements are contained in the same family, there exists a corresponding set. This set is constructed by the function Set (Reference: Set) which takes the list as its argument and returns a set obtained from this list by ignoring holes and duplicates and by sorting the elements.
The elements of the sets used in the examples of this section are strings.
gap> fruits:= ["apple", "strawberry", "cherry", "plum"]; [ "apple", "strawberry", "cherry", "plum" ] gap> IsSSortedList(fruits); false gap> fruits:= Set(fruits); [ "apple", "cherry", "plum", "strawberry" ]
Note that the original list fruits is not changed by the function Set (Reference: Set). We have to make a new assignment to the variable fruits in order to make it a set.
The operator in is used to test whether an object is an element of a set. It returns a boolean value true or false.
gap> "apple" in fruits; true gap> "banana" in fruits; false
The operator in can also be applied to ordinary lists. It is however much faster to perform a membership test for sets since sets are always sorted and a binary search can be used instead of a linear search. New elements may be added to a set by the function AddSet (Reference: AddSet) which takes the set fruits as its first argument and an element as its second argument and adds the element to the set if it wasn't already there. Note that the object fruits is changed.
gap> AddSet(fruits, "banana"); gap> fruits; # The banana is inserted in the right place. [ "apple", "banana", "cherry", "plum", "strawberry" ] gap> AddSet(fruits, "apple"); gap> fruits; # fruits has not changed. [ "apple", "banana", "cherry", "plum", "strawberry" ]
Note that inserting new elements into a set with AddSet (Reference: AddSet) is usually more expensive than simply adding new elements at the end of a list.
Sets can be intersected by the function Intersection (Reference: Intersection) and united by the function Union (Reference: Union) which both take two sets as their arguments and return the intersection resp. union of the two sets as a new object.
gap> breakfast:= ["tea", "apple", "egg"]; [ "tea", "apple", "egg" ] gap> Intersection(breakfast, fruits); [ "apple" ]
The arguments of the functions Intersection (Reference: Intersection) and Union (Reference: Union) could be ordinary lists, while their result is always a set. Note that in the preceding example at least one argument of Intersection (Reference: Intersection) was not a set. The functions IntersectSet (Reference: IntersectSet) and UniteSet (Reference: UniteSet) also form the intersection resp. union of two sets. They will however not return the result but change their first argument to be the result. Try them carefully.
A range is a finite arithmetic progression of integers. This is another special kind of list. A range is described by the first two values and the last value of the arithmetic progression which are given in the form [first,second..last]. In the usual case of an ascending list of consecutive integers the second entry may be omitted.
gap> [1..999999]; # a range of almost a million numbers [ 1 .. 999999 ] gap> [1, 2..999999]; # this is equivalent [ 1 .. 999999 ] gap> [1, 3..999999]; # here the step is 2 [ 1, 3 .. 999999 ] gap> Length( last ); 500000 gap> [ 999999, 999997 .. 1 ]; [ 999999, 999997 .. 1 ]
This compact printed representation of a fairly long list corresponds to a compact internal representation. The function IsRange (Reference: IsRange) tests whether an object is a range, the function ConvertToRangeRep (Reference: ConvertToRangeRep) changes the representation of a list that is in fact a range to this compact internal representation.
gap> a:= [-2,-1,0,1,2,3,4,5]; [ -2, -1, 0, 1, 2, 3, 4, 5 ] gap> IsRange( a ); true gap> ConvertToRangeRep( a );; a; [ -2 .. 5 ] gap> a[1]:= 0;; IsRange( a ); false
Note that this change of representation does not change the value of the list a. The list a still behaves in any context in the same way as it would have in the long representation.
Given a list pp of permutations we can form their product by means of a for loop instead of writing down the product explicitly.
gap> pp:= [ (1,3,2,6,8)(4,5,9), (1,6)(2,7,8), (1,5,7)(2,3,8,6), > (1,8,9)(2,3,5,6,4), (1,9,8,6,3,4,7,2)];; gap> prod:= (); () gap> for p in pp do > prod:= prod*p; > od; gap> prod; (1,8,4,2,3,6,5,9)
First a new variable prod is initialized to the identity permutation (). Then the loop variable p takes as its value one permutation after the other from the list pp and is multiplied with the present value of prod resulting in a new value which is then assigned to prod.
The for loop has the following syntax
for var in list do statements od;
The effect of the for loop is to execute the statements for every element of the list. A for loop is a statement and therefore terminated by a semicolon. The list of statements is enclosed by the keywords do and od (reverse do). A for loop returns no value. Therefore we had to ask explicitly for the value of prod in the preceding example.
The for loop can loop over any kind of list, even a list with holes. In many programming languages the for loop has the form
for var from first to last do statements od;
In GAP this is merely a special case of the general for loop as defined above where the list in the loop body is a range (see 3.5):
for var in [first..last] do statements od;
You can for instance loop over a range to compute the factorial \(15!\) of the number \(15\) in the following way.
gap> ff:= 1; 1 gap> for i in [1..15] do > ff:= ff * i; > od; gap> ff; 1307674368000
The while loop has the following syntax
while condition do statements od;
The while loop loops over the statements as long as the condition evaluates to true. Like the for loop the while loop is terminated by the keyword od followed by a semicolon.
We can use our list primes to perform a very simple factorization. We begin by initializing a list factors to the empty list. In this list we want to collect the prime factors of the number 1333. Remember that a list has to exist before any values can be assigned to positions of the list. Then we will loop over the list primes and test for each prime whether it divides the number. If it does we will divide the number by that prime, add it to the list factors and continue.
gap> n:= 1333;; gap> factors:= [];; gap> for p in primes do > while n mod p = 0 do > n:= n/p; > Add(factors, p); > od; > od; gap> factors; [ 31, 43 ] gap> n; 1
As n now has the value 1 all prime factors of 1333 have been found and factors contains a complete factorization of 1333. This can of course be verified by multiplying 31 and 43.
This loop may be applied to arbitrary numbers in order to find prime factors. But as primes is not a complete list of all primes this loop may fail to find all prime factors of a number greater than 2000, say. You can try to improve it in such a way that new primes are added to the list primes if needed.
You have already seen that list objects may be changed. This of course also holds for the list in a loop body. In most cases you have to be careful not to change this list, but there are situations where this is quite useful. The following example shows a quick way to determine the primes smaller than 1000 by a sieve method. Here we will make use of the function Unbind to delete entries from a list, and the if statement covered in 4.2.
gap> primes:= [];; gap> numbers:= [2..1000];; gap> for p in numbers do > Add(primes, p); > for n in numbers do > if n mod p = 0 then > Unbind(numbers[n-1]); > fi; > od; > od;
The inner loop removes all entries from numbers that are divisible by the last detected prime p. This is done by the function Unbind which deletes the binding of the list position numbers[n-1] to the value n so that afterwards numbers[n-1] no longer has an assigned value. The next element encountered in numbers by the outer loop necessarily is the next prime.
In a similar way it is possible to enlarge the list which is looped over. This yields a nice and short orbit algorithm for the action of a group, for example.
More about for and while loops can be found in the sections Reference: While and Reference: For.
There is a more comfortable way than that given in the previous section to compute the product of a list of numbers or permutations.
gap> Product([1..15]); 1307674368000 gap> Product(pp); (1,8,4,2,3,6,5,9)
The function Product (Reference: Product) takes a list as its argument and computes the product of the elements of the list. This is possible whenever a multiplication of the elements of the list is defined. So Product (Reference: Product) executes a loop over all elements of the list.
There are other often used loops available as functions. Guess what the function Sum (Reference: Sum) does. The function List (Reference: Lists) may take a list and a function as its arguments. It will then apply the function to each element of the list and return the corresponding list of results. A list of cubes is produced as follows with the function cubed from Section 4.
gap> cubed:= x -> x^3;; gap> List([2..10], cubed); [ 8, 27, 64, 125, 216, 343, 512, 729, 1000 ]
To add all these cubes we might apply the function Sum (Reference: Sum) to the last list. But we may as well give the function cubed to Sum (Reference: Sum) as an additional argument.
gap> Sum(last) = Sum([2..10], cubed); true
The primes less than 30 can be retrieved out of the list primes from Section 3.1 by the function Filtered (Reference: Filtered). This function takes the list primes and a property as its arguments and will return the list of those elements of primes which have this property. Such a property will be represented by a function that returns a boolean value. In this example the property of being less than 30 can be represented by the function x -> x < 30 since x < 30 will evaluate to true for values x less than 30 and to false otherwise.
gap> Filtered(primes, x -> x < 30); [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ]
We have already mentioned the operator { } that forms sublists. It takes a list of positions as its argument and will return the list of elements from the original list corresponding to these positions.
gap> primes{ [1 .. 10] }; [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ]
Finally we mention the function ForAll (Reference: ForAll) that checks whether a property holds for all elements of a list. It takes as its arguments a list and a function that returns a boolean value. ForAll (Reference: ForAll) checks whether the function returns true for all elements of the list.
gap> list:= [ 1, 2, 3, 4 ];; gap> ForAll( list, x -> x > 0 ); true gap> ForAll( list, x -> x in primes ); false
You will find more predefined for loops in chapter Reference: Lists.
This section describes how GAP uses lists to represent row vectors and matrices. A row vector is a dense list of elements from a common field. A matrix is a dense list of row vectors over a common field and of equal length.
gap> v:= [3, 6, 2, 5/2];; IsRowVector(v); true
Row vectors may be added and multiplied by scalars from their field. Multiplication of row vectors of equal length results in their scalar product.
gap> 2 * v; v * 1/3; [ 6, 12, 4, 5 ] [ 1, 2, 2/3, 5/6 ] gap> v * v; # the scalar product of `v' with itself 221/4
Note that the expression v * 1/3 is actually evaluated by first multiplying v by 1 (which yields again v) and by then dividing by 3. This is also an allowed scalar operation. The expression v/3 would result in the same value.
Such arithmetical operations (if the results are again vectors) result in mutable vectors except if the operation is binary and both operands are immutable; thus the vectors shown in the examples above are all mutable.
So if you want to produce a mutable list with 100 entries equal to 25, you can simply say 25 + 0 * [ 1 .. 100 ]. Note that ranges are also vectors (over the rationals), and that [ 1 .. 100 ] is mutable.
A matrix is a dense list of row vectors of equal length.
gap> m:= [[1,-1, 1], > [2, 0,-1], > [1, 1, 1]]; [ [ 1, -1, 1 ], [ 2, 0, -1 ], [ 1, 1, 1 ] ] gap> m[2][1]; 2
Syntactically a matrix is a list of lists. So the number 2 in the second row and the first column of the matrix m is referred to as the first element of the second element of the list m via m[2][1].
A matrix may be multiplied by scalars, row vectors and other matrices. (If the row vectors and matrices involved in such a multiplication do not have suitable dimensions then the "missing" entries are treated as zeros, so the results may look unexpectedly in such cases.)
gap> [1, 0, 0] * m; [ 1, -1, 1 ] gap> [1, 0, 0, 2] * m; [ 1, -1, 1 ] gap> m * [1, 0, 0]; [ 1, 2, 1 ] gap> m * [1, 0, 0, 2]; [ 1, 2, 1 ]
Note that multiplication of a row vector with a matrix will result in a linear combination of the rows of the matrix, while multiplication of a matrix with a row vector results in a linear combination of the columns of the matrix. In the latter case the row vector is considered as a column vector.
A vector or matrix of integers can also be multiplied with a finite field scalar and vice versa. Such products result in a matrix over the finite field with the integers mapped into the finite field in the obvious way. Finite field matrices are nicer to read when they are Displayed rather than Printed. (Here we write Z(q) to denote a primitive root of the finite field with q elements.)
gap> Display( m * One( GF(5) ) ); 1 4 1 2 . 4 1 1 1 gap> Display( m^2 * Z(2) + m * Z(4) ); z = Z(4) z^1 z^1 z^2 1 1 z^2 z^1 z^1 z^2
Submatrices can easily be extracted using the expression mat{rows}{columns}. They can also be assigned to, provided the big matrix is mutable (which it is not if it is the result of an arithmetical operation, see above).
gap> sm := m{ [ 1, 2 ] }{ [ 2, 3 ] }; [ [ -1, 1 ], [ 0, -1 ] ] gap> sm{ [ 1, 2 ] }{ [2] } := [[-2],[0]];; sm; [ [ -1, -2 ], [ 0, 0 ] ]
The first curly brackets contain the selection of rows, the second that of columns.
Matrices appear not only in linear algebra, but also as group elements, provided they are invertible. Here we have the opportunity to meet a group-theoretical function, namely Order (Reference: Order), which computes the order of a group element.
gap> Order( m * One( GF(5) ) ); 8 gap> Order( m ); infinity
For matrices whose entries are more complex objects, for example rational functions, GAP's Order (Reference: Order) methods might not be able to prove that the matrix has infinite order, and one gets the following warning.
#I Order: warning, order of <mat> might be infinite
In such a case, if the order of the matrix really is infinite, you will have to interrupt GAP by pressing ctl-C (followed by ctl-D or quit; to leave the break loop).
To prove that the order of m is infinite, we also could look at the minimal polynomial of m over the rationals.
gap> f:= MinimalPolynomial( Rationals, m );; Factors( f ); [ x_1-2, x_1^2+3 ]
Factors (Reference: Factors) returns a list of irreducible factors of the polynomial f. The first irreducible factor \(X-2\) reveals that 2 is an eigenvalue of m, hence its order cannot be finite.
A record provides another way to build new data structures. Like a list a record contains subobjects. In a record the elements, the so-called record components, are not indexed by numbers but by names.
In this section you will see how to define and how to use records. Records are changed by assignments to record components or by unbinding record components.
Initially a record is defined as a comma separated list of assignments to its record components.
gap> date:= rec(year:= 1997, > month:= "Jul", > day:= 14); rec( day := 14, month := "Jul", year := 1997 )
The value of a record component is accessible by the record name and the record component name separated by one dot as the record component selector.
gap> date.year; 1997
Assignments to new record components are possible in the same way. The record is automatically resized to hold the new component.
gap> date.time:= rec(hour:= 19, minute:= 23, second:= 12); rec( hour := 19, minute := 23, second := 12 ) gap> date; rec( day := 14, month := "Jul", time := rec( hour := 19, minute := 23, second := 12 ), year := 1997 )
Records are objects that may be changed. An assignment to a record component changes the original object. The remarks made in Sections 3.2 and 3.3 about identity and mutability of lists are also true for records.
Sometimes it is interesting to know which components of a certain record are bound. This information is available from the function RecNames (Reference: RecNames), which takes a record as its argument and returns a list of names of all bound components of this record as a list of strings.
gap> RecNames(date); [ "time", "year", "month", "day" ]
Now return to Sections 3.2 and 3.3 and find out what these sections mean for records.
(The following cross-references point to the GAP Reference Manual.)
You will find more about lists, sets, and ranges in Chapter Reference: Lists, in particular more about identical lists in Section Reference: Identical Lists. A more detailed description of strings is contained in Chapter Reference: Strings and Characters. Fields are described in Chapter Reference: Fields and Division Rings, some known fields in GAP are described in Chapters Reference: Rational Numbers, Reference: Abelian Number Fields, and Reference: Finite Fields. Row vectors and matrices are described in more detail in Chapters Reference: Row Vectors and Reference: Matrices. Vector spaces are described in Chapter Reference: Vector Spaces, further matrix related structures are described in Chapters Reference: Matrix Groups, Reference: Algebras, and Reference: Lie Algebras.
You will find more list operations in Chapter Reference: Lists.
Records and functions for records are described in detail in Chapter Reference: Records.
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