This chapter contains an overview of most important changes introduced in **GAP** 4.7.2 release (the first public release of **GAP** 4.7). It also contains information about subsequent update releases for **GAP** 4.7.

Improved and extended functionality:

The methods for computing conjugacy classes of permutation groups have been rewritten from scratch to enable potential use for groups in different representations. As a byproduct the resulting code is (sometimes notably) faster. It also now is possible to calculate canonical conjugacy class representatives in permutation groups, which can be beneficial when calculating character tables.

The methods for determining (conjugacy classes of) subgroups in non-solvable groups have been substantially improved in speed and scope for groups with multiple nonabelian composition factors.

There is a new method for calculating the maximal subgroups of a permutation group (with chief factors of width less or equal 5) without calculating the whole subgroup lattice.

If available, information from the table of marks library is used to speed up subgroup calculations in almost simple factor groups.

The broader availability of maximal subgroups is used to improve the calculation of double cosets.

To illustrate the improvements listed above, one could try, for example

g:=WreathProduct(MathieuGroup(11),Group((1,2))); Length(ConjugacyClassesSubgroups(g));

and

g:=SemidirectProduct(GL(3,5),GF(5)^3); g:=Image(IsomorphismPermGroup(g)); MaximalSubgroupClassReps(g);

Computing the exponent of a finite group \(G\) could be extremely slow. This was due to a slow default method being used, which computed all conjugacy classes of elements in order to compute the exponent. We now instead compute Sylow subgroups \(P_1\), ..., \(P_k\) of \(G\) and use the easily verified equality \(exp(G) = exp(P_1) x ... x exp(P_k)\). This is usually at least as fast and in many cases orders of magnitude faster.

gap> G:=SmallGroup(2^7*9,33);; gap> H:=DirectProduct(G, ElementaryAbelianGroup(2^10));; gap> Exponent(H); # should take at most a few milliseconds 72 gap> K := PerfectGroup(2688,3);; gap> Exponent(K); # should take at most a few seconds 168

The functionality in

**GAP**for transformations and transformation semigroups has been rewritten and extended. Partial permutations and inverse semigroups have been newly implemented. The documentation for transformations and transformation semigroups has been improved. Transformations and partial permutations are implemented in the**GAP**kernel. Methods for calculating attributes of transformations and partial permutations, and taking products, and so are also implemented in the kernel. The new implementations are largely backwards compatible; some exceptions are given below.The degree of a transformation

`f`

is usually defined as the largest positive integer where`f`

is defined. In previous versions of**GAP**, transformations were only defined on positive integers less than their degree, it was only possible to multiply transformations of equal degree, and a transformation did not act on any point exceeding its degree. Starting with**GAP**4.7, transformations behave more like permutations, in that they fix unspecified points and it is possible to multiply arbitrary transformations.in the display of a transformation, the trailing fixed points are no longer printed. More precisely, in the display of a transformation

`f`

if`n`

is the largest value such that`n^f<>n`

or`i^f=n`

for some`i<>n`

, then the values exceeding`n`

are not printed.the display for semigroups of transformations now includes more information, for example

`<transformation semigroup on 10 pts with 10 generators>`

and`<inverse partial perm semigroup on 10 pts with 10 generators>`

.transformations which define a permutation can be inverted, and groups of transformations can be created.

Further information regarding transformations and partial permutations, can be found in the relevant chapters of the reference manual.

The code for Rees matrix semigroups has been completely rewritten to fix the numerous bugs in the previous versions. The display of a Rees matrix semigroup has also been improved to include the numbers of rows and columns, and the underlying semigroup. Again the new implementations should be backwards compatible with the exception that the display is different.

The code for magmas with a zero adjoined has been improved so that it is possible to access more information about the original magma. The display has also been changed to indicate that the created magma is a magma with zero adjoined (incorporating the display of the underlying magma). Elements of a magma with zero are also printed so that it is clear that they belong to a magma with zero.

If a semigroup is created by generators in the category IsMultiplicativeElementWithOneCollection and CanEasilyCompareElements, then it is now checked if the One of the generators is given as a generator. In this case, the semigroup is created as a monoid.

Added a new operation

`GrowthFunctionOfGroup`

(Reference: GrowthFunctionOfGroup) that gives sizes of distance spheres in the Cayley graph of a group.A new group constructor

`FreeAbelianGroup`

(Reference: FreeAbelianGroup) for free abelian groups has been added. By default, it creates suitable fp groups. Though free abelian groups groups do not offer much functionality right now, in the future other implementations may be provided, e.g. by the**Polycyclic**package.The message about halving the pool size at startup is only shown when

`-D`

command line option is used (see Reference: Command Line Options). [Suggested by Volker Braun]An info class called

`InfoObsolete`

(Reference: InfoObsolete) with the default level 0 is introduced. Setting it to 1 will trigger warnings at runtime if an obsolete variable declared with`DeclareObsoleteSynonym`

is used. This is recommended for testing**GAP**distribution and packages.The

**GAP**help system now recognises some common different spelling patterns (for example, -ise/-ize, -isation/-ization, solvable/soluble) and searches for all possible spelling options even when the synonyms are not declared.Added new function

`Cite`

(Reference: Cite) which produces citation samples for**GAP**and packages.It is now possible to compile

**GAP**with user-supplied`CFLAGS`

which now will not be overwritten by**GAP**default settings. [Suggested by Jeroen Demeyer]

Fixed bugs:

`Union`

(Reference: Union) had \(O(n^3)\) behaviour when given many ranges (e.g. it could take 10 seconds to find a union of 1000 1-element sets). The new implementation reduces that to \(O(n log n)\) (and 4ms for the 10 second example), at the cost of not merging ranges as well as before in some rare cases.`IsLatticeOrderBinaryRelation`

only checked the existence of upper bounds but not the uniqueness of the least upper bound (and dually for lower bounds), so in some cases it could return the wrong answer. [Reported by Attila Egri-Nagy]`LowIndexSubgroupsFpGroup`

(Reference: LowIndexSubgroupsFpGroup) triggered a break loop if the list of generators of the 2nd argument contained the identity element of the group. [Reported by Ignat Soroko]Fixed regression in heuristics used by

`NaturalHomomorphismByNormalSubgroup`

(Reference: NaturalHomomorphismByNormalSubgroup) that could produce a permutation representation of an unreasonably large degree. [Reported by Izumi Miyamoto]Fixed inconsistent behaviour of

`QuotientMod( Integers, r, s, m )`

in the case where`s`and`m`are not coprime. This fix also corrects the division behaviour of`ZmodnZ`

objects, see`QuotientMod`

(Reference: QuotientMod) and`ZmodnZ`

(Reference: ZmodnZ). [Reported by Mark Dickinson]Fixed an oversight in the loading process causing

`OnQuit`

(Reference: OnQuit) not resetting the options stack after exiting the break loop.Empty strings were treated slightly differently than other strings in the

**GAP**kernel, for historical reasons. This resulted in various inconsistencies. For example,`IsStringRep("")`

returned true, but a method installed for arguments of type`IsStringRep`

would NOT be invoked when called with an empty string.We remove this special case in the

**GAP**kernel (which dates back the very early days of**GAP**4 in 1996). This uncovered one issue in the kernel function`POSITION_SUBSTRING`

(when calling it with an empty string as second argument), which was also fixed.The parser for floating point numbers contained a bug that could cause

**GAP**to crash or to get into a state where the only action left to the user was to exit**GAP**via Ctrl-D. For example, entering four dots with spaces between them on the**GAP**prompt and then pressing the return key caused**GAP**to exit.The reason was (ironically) an error check in the innards of the float parser code which invoked the

**GAP**`Error()`

function at a point where it should not have.Removing the last character in a string was supposed to overwrite the old removed character in memory with a zero byte, but failed to do so due to an off-by-one error. For most

**GAP**operations, this has no visible effect, except for those which directly operate on the underlying memory representation of strings. For example, when trying to use such a string to reference a record entry, a (strange) error could be triggered.`ViewString`

(Reference: ViewString) and`DisplayString`

(Reference: DisplayString) are now handling strings, characters and immediate FFEs in a consistent manner.Multiple fixes to the build process for less common Debian platforms (arm, ia64, mips, sparc, GNU/Hurd). [Suggested by Bill Allombert]

Fixes for several regressions in the

`gac`

script. [Suggested by Bill Allombert]

Changed functionality:

It is not possible now to call

`WreathProduct`

(Reference: WreathProduct) with 2nd argument`H`not being a permutation group, without using the 3rd argument specifying the permutation representation. This is an incompatible change but it will produce an error instead of a wrong result. The former behaviour of`WreathProduct`

(Reference: WreathProduct) may now be achieved by using`StandardWreathProduct`

(Reference: StandardWreathProduct) which returns the wreath product for the (right regular) permutation action of`H`on its elements.The function

`ViewLength`

to specify the maximal number of lines that are printed in`ViewObj`

(Reference: ViewObj) became obsolete, since there was already a user preference`ViewLength`

to specify this. The value of this preference is also accessible in`GAPInfo.ViewLength`

.

At the time of the release of **GAP** 4.6.5 there were 107 packages redistributed with **GAP**. The first public release of **GAP** 4.7 contains 114 packages.

One of essential changes is that the **Citrus** package by J.Mitchell has been renamed to **Semigroups**. The package has been completely overhauled, the performance has been improved, and the code has been generalized so that in the future the same code can be used to compute with other types of semigroups.

Furthermore, new packages that have been added to the redistribution since the release of **GAP** 4.6.5 are:

**4ti2interface**package by Sebastian Gutsche, providing an interface to**4ti2**, a software package for algebraic, geometric and combinatorial problems on linear spaces (http://www.4ti2.de).**CoReLG**by Heiko Dietrich, Paolo Faccin and Willem de Graaf for calculations in real semisimple Lie algebras.**IntPic**package by Manuel Delgado, aimed at providing a simple way of getting a pictorial view of sets of integers. The main goal of the package is producing**Tikz**code for arrays of integers. The code produced is to be included in a LaTeX file, which can then be processed. Some of the integers are emphasized by using different colors for the cells containing them.**LieRing**by Serena Cicalo and Willem de Graaf for constructing finitely-presented Lie rings and calculating the Lazard correspondence. The package also provides a database of small \(n\)-Engel Lie rings.**LiePRing**package by Michael Vaughan-Lee and Bettina Eick, introducing a new datastructure for nilpotent Lie rings of prime-power order. This allows to define such Lie rings for specific primes as well as for symbolic primes and other symbolic parameters. The package also includes a database of nilpotent Lie rings of order at most \(p^7\) for all primes \(p > 3\).**ModIsom**by Bettina Eick, which contains various methods for computing with nilpotent associative algebras. In particular, it contains a method to determine the automorphism group and to test isomorphisms of such algebras over finite fields and of modular group algebras of finite \(p\)-groups. Further, it contains a nilpotent quotient algorithm for finitely presented associative algebras and a method to determine Kurosh algebras.**SLA**by Willem de Graaf for computations with simple Lie algebras. The main topics of the package are nilpotent orbits, theta-groups and semisimple subalgebras.

Furthermore, some packages have been upgraded substantially since the **GAP** 4.6.5 release:

**ANUPQ**package by Greg Gamble, Werner Nickel and Eamonn O'Brien has been updated after Max Horn joined it as a maintainer. As a result, it is now much easier to install and use it with the current**GAP**release.**Wedderga**package by Osnel Broche Cristo, Allen Herman, Alexander Konovalov, Aurora Olivieri, Gabriela Olteanu, Ángel del Río and Inneke Van Gelder has been extended to include functions for calculating local and global Schur indices of ordinary irreducible characters of finite groups, cyclotomic algebras over abelian number fields, and rational quaternion algebras (contribution by Allen Herman).

Fixed bugs which could lead to incorrect results:

Incorrect result returned by

`AutomorphismGroup(PSp(4,2^n))`

. [Reported by Anvita]The

`Order`

(Reference: Order) method for group homomorphisms newly introduced in**GAP**4.7 had a bug that caused it to sometimes return incorrect results. [Reported by Benjamin Sambale]

Fixed bugs that could lead to break loops:

Several bugs were fixed and missing methods were introduced in the new code for transformations, partial permutations and semigroups that was first included in

**GAP**4.7. Some minor corrections were made in the documentation for transformations.Break loop in

`IsomorphismFpMonoid`

when prefixes in generators names were longer than one letter. [Reported by Dmytro Savchuk and Yevgen Muntyan]Break loop while displaying the result of

`MagmaWithInversesByMultiplicationTable`

(Reference: MagmaWithInversesByMultiplicationTable). [Reported by Grahame Erskine]

Improved functionality:

Better detection of UTF-8 terminal encoding on some systems. [Suggested by Andries Brouwer]

This release was prepared immediately after **GAP** 4.7.3 to revert the fix of the error handling for the single quote at the end of an input line, contained in **GAP** 4.7.3. It happened that (only on Windows) the fix caused error messages in one of the packages.

Fixed bugs which could lead to incorrect results:

`InstallValue`

(Reference: InstallValue) cannot handle immediate values, characters or booleans for technical reasons. A check for such values was introduced to trigger an error message and prevent incorrect results caused by this. [Reported by Sebastian Gutsche]`KnowsDictionary`

(Reference: KnowsDictionary) and`LookupDictionary`

(Reference: LookupDictionary) methods for`IsListLookupDictionary`

were using`PositionFirstComponent`

(Reference: PositionFirstComponent); the latter is only valid on sorted lists, but in`IsListLookupDictionary`

the underlying list is NOT sorted in general, leading to bogus results.

Other fixed bugs:

A bug in

`DirectProductElementsFamily`

which used`CanEasilyCompareElements`

(Reference: CanEasilyCompareElements) instead of`CanEasilySortElements`

(Reference: CanEasilySortElements).Fixed wrong

`Infolevel`

message that caused a break loop for some automorphism group computations.Fixed an error that sometimes caused a break loop in

`HallSubgroup`

(Reference: HallSubgroup). [Reported by Benjamin Sambale]Fixed a rare error in computation of conjugacy classes of a finite group by homomorphic images, providing fallback to a default algorithm.

Fixed an error in the calculation of Frattini subgroup in the case of the trivial radical.

Several minor bugs were fixed in the documentation, kernel, and library code for transformations.

Fixed errors in

`NumberPerfectGroups`

(Reference: NumberPerfectGroups) and`NumberPerfectLibraryGroups`

(Reference: NumberPerfectLibraryGroups) not being aware that there are no perfect groups of odd order.Restored the ability to build

**GAP**on OS X 10.4 and 10.5 which was accidentally broken in the previous**GAP**release by using the build option not supported by these versions.Fixed some problems for ia64 and sparc architectures. [Reported by Bill Allombert and Volker Braun]

New package added for the redistribution with **GAP**:

**permut**package by A.Ballester-Bolinches, E.Cosme-Llópez, and R.Esteban-Romero to deal with permutability in finite groups.

Fixed bugs which could lead to incorrect results:

A bug that may cause

`ShortestVectors`

(Reference: ShortestVectors) to return an incomplete list. [Reported by Florian Beye]A bug that may lead to incorrect results and infinite loops when

**GAP**is compiled without GMP support using gcc 4.9.A bug that may cause

`OrthogonalEmbeddings`

(Reference: OrthogonalEmbeddings) to return an incomplete result. [Reported by Benjamin Sambale]

Fixed bugs that could lead to break loops:

`ClosureGroup`

(Reference: ClosureGroup) should be used instead of`ClosureSubgroup`

(Reference: ClosureSubgroup) in case there is no parent group, otherwise some calculations such as e.g.`NormalSubgroups`

(Reference: NormalSubgroups) may fail. [Reported by Dmitrii Pasechnik]Fixed a line in the code that used a hard-coded identity permutation, not a generic identity element of a group. [Reported by Toshio Sumi]

Fixed a problem in the new code for calculating maximal subgroups that caused a break loop for some groups from the transitive groups library. [Reported by Petr Savicky]

Fixed a problem in

`ClosureSubgroup`

(Reference: ClosureSubgroup) not accepting some groups without`Parent`

(Reference: Parent). [Reported by Inneke van Gelder]

Other fixed bugs:

Eliminated a number of compiler warnings detected with some newer versions of

**C**compilers.Some minor bugs in the transformation and partial permutation code and documentation were resolved.

New features:

Introduced some arithmetic operations for infinity and negative infinity, see Reference: infinity.

Introduced new property

`IsGeneratorsOfSemigroup`

(Reference: IsGeneratorsOfSemigroup) which reflects wheter the list or collection generates a semigroup.

Fixed bugs which could lead to incorrect results:

Fixed a bug in

`Union`

(Reference: Union) (actually, in the internal library function`JoinRanges`

) caused by downward running ranges. [Reported by Matt Fayers]Fixed a bug where recursive records might be printed with the wrong component name, coming from component names being ordered differently in two different pieces of code. [Reported by Thomas Breuer]

The usage of

`abs`

in`src/gmpints.c`

was replaced by`AbsInt`

. The former is defined to operate on 32-bit integers even if**GAP**is compiled in 64-bit mode. That lead to truncating**GAP**integers and caused a crash in`RemInt`

(Reference: RemInt), reported by Willem De Graaf and Heiko Dietrich. Using`AbsInt`

fixes the crash, and ensures the correct behaviour on 32-bit and 64-bit builds.

Fixed bugs that could lead to break loops:

A problem with

`ProbabilityShapes`

(Reference: ProbabilityShapes) not setting frequencies list for small degrees. [Reported by Daniel Błażewicz and independently by Mathieu Gagne]An error when generating a free monoid of rank infinity. [Reported by Nick Loughlin]

Several bugs with the code for Rees matrix semigroups not handling trivial cases properly.

A bug in

`IsomorphismTypeInfoFiniteSimpleGroup`

(Reference: IsomorphismTypeInfoFiniteSimpleGroup) affecting one particular group due to a misformatting in a routine that translates between the Chevalley type and the name used in the table (in this case,`"T"`

was used instead of`["T"]`

). [Reported by Petr Savicky]

Other fixed bugs:

The

`Basis`

(Reference: Basis) method for full homomorphism spaces of linear mappings did not set basis vectors which could be obtained by`GeneratorsOfLeftModule`

(Reference: GeneratorsOfLeftModule).A problem with

`GaloisType`

(Reference: GaloisType) entering an infinite loop in the routine for approximating a root. [Reported by Daniel Błażewicz]Fixed the crash when

**GAP**is called when the environment variables`HOME`

or`PATH`

are unset. [Reported by Bill Allombert]

Furthermore, new packages that have been added to the redistribution since the release of **GAP** 4.7.6 are:

**json**package by Christopher Jefferson, providing a mapping between the**JSON**markup language and**GAP****SglPPow**package by Bettina Eick and Michael Vaughan-Lee, providing the database of \(p\)-groups of order \(p^7\) for \(p > 11\), and of order \(3^8\).

Fixed bugs which could lead to incorrect results:

Added two groups of degree 1575 which were missing in the library of first primitive groups. [Reported by Gordon Royle]

Fixed the error in the code for algebra module elements in packed representation caused by the use of

`Objectify`

(Reference: Objectify) with the type of the given object instead of`ObjByExtRep`

(Reference: ObjByExtRep) as recommended in Reference: Further Improvements in Implementing Residue Class Rings. The problem was that after calculating`u+v`

where one of the summands was known to be zero, this knowledge was wrongly passed to the sum via the type. [Reported by Istvan Szollosi]Fixed a bug in

`PowerMod`

(Reference: PowerMod) causing wrong results for univariate Laurent polynomials when the two polynomial arguments are stored with the same non-zero shift. [Reported by Max Horn]

Furthermore, new packages that have been added to the redistribution since the release of **GAP** 4.7.7 are:

**PatternClass**by Michael Albert, Ruth Hoffmann and Steve Linton, allowing to explore the permutation pattern classes build by token passing networks. Amongst other things, it can compute the basis of a permutation pattern class, create automata from token passing networks and check if the deterministic automaton is a possible representative of a token passing network.**QPA**by Edward Green and Øyvind Solberg, providing data structures and algorithms for computations with finite dimensional quotients of path algebras, and with finitely generated modules over such algebras. It implements data structures for quivers, quotients of path algebras, and modules, homomorphisms and complexes of modules over quotients of path algebras.

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