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#! @Chapter The &SOTGrps; package
#! With some overlaps, the &SOTGrps; package extends the Small Group Library to give access to some more
#! <Q>small</Q> orders. For example, it constructs a
#! complete and irredundant list of isomorphism type representatives of the groups of order
#! - that factorises into at most four primes;
#! - <M>p^4q</M>, for distinct primes <M>p</M> and <M>q</M>.
#!
#! The mathematical background for this package is described in <Cite Key="DEP22"/>.
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#! @Section Main functions
#!
#! In addition to the functions described below, the &SOTGrps; package also extends the
#! the Small Groups Library as provided by the &SmallGrp; package: with &SOTGrps; loaded,
#! functions such as <C>NumberSmallGroups</C>, <C>SmallGroup</C> or <C>IdGroup</C>
#! will work for orders support by &SOTGrps; but not by &SmallGrp;.
#!
#! Note: for orders support by &SOTGrps; *and* by &SmallGrp;, the respective ids as
#! produced by <C>IdGroup</C> versus <C>IdSOTGroup</C> in general do not agree.
#! In a future version we may provided functions to convert between them.
#! @Description
#! takes in a number <A>n</A> that factorises into at most four primes or is of the form <M>p^4q</M> (<M>p</M>, <M>q</M> are distinct primes),
#! and returns a complete and duplicate-free list of isomorphism class representatives of the groups of order <A>n</A>.
#! Solvable groups are using refined polycyclic presentations.
#! By default, solvable groups are constructed in the filter <C>IsPcGroup</C>,
#! but if the optional argument <A>filter</A> is set to <C>IsPcpGroup</C> then
#! the groups are constructed in that filter instead.
#! Nonsolvable groups are always returned as permutation groups.
#! @Arguments n [, filter]
#! @BeginExampleSession
#! gap> AllSOTGroups(60);
#! [ <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,
#! <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,
#! <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,
#! <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,
#! <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,
#! <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,
#! Alt( [ 1 .. 5 ] ) ]
#! @EndExampleSession
DeclareGlobalFunction("AllSOTGroups");
#! @Description
#! takes in a number <A>n</A> that factorises into at most four primes or of the form <M>p^4q</M> (<M>p</M>, <M>q</M> are distinct primes),
#! and returns the number of isomorphism types of groups of order <A>n</A>.
#! @Arguments n
#! @BeginExampleSession
#! gap> NumberOfSOTGroups(2*3*5*7);
#! 12
#! gap> NumberOfSOTGroups(2*3*5*7*11);
#! Error, Order 2310 is not supported by SOTGrps.
#! Please refer to the SOTGrps documentation for the list of supported orders.
#! @EndExampleSession
DeclareGlobalFunction("NumberOfSOTGroups");
#! @Description
#! takes in a pair of numbers <A>n, i</A>, where <A>n</A> factorises into at most four primes or of the form <M>p^4q</M> (<M>p</M>, <M>q</M> are distinct primes),
#! and returns the <A>i</A>-th group with respect to the ordering of
#! the list <C>AllSOTGroups(<A>n</A>)</C> without constructing all groups in the list.
#! The option of constructing a PcpGroup is available for solvable groups.
#! @Arguments n, i[, arg]
#! @BeginExampleSession
#! gap> SOTGroup(2*3*5*7, 1);
#! <pc group of size 210 with 4 generators>
#! @EndExampleSession
#! If the input <A>i</A> exceeds the number of groups of order <A>n</A>, an error message is returned.
DeclareGlobalFunction("SOTGroup");
#! @Description
#! takes in a group of order determines the SOT library number of <A>G</A>;
#! that is, the function returns a pair [<A>n</A>, <A>i</A>] where <A>G</A> is isomorphic to <C>SOTGroup(<A>n</A>,<A>i</A>)</C>.
#! Note that if the input group is a PcpGroup, this may result in slow runtime, as <C>IdSOTGroup</C> may compute the <C>Centre</C> and/or the <C>FittingSubgroup</C>,
#! which is slow for PcpGroups.
#! @Arguments G
DeclareAttribute( "IdSOTGroup", IsGroup );
#! @Description
#! determines whether two groups <A>G</A>, <A>H</A> are isomorphic. It is assumed that the input groups are available in the &SOTGrps; library.
#! @Arguments G, H
#! @BeginExampleSession
#! gap> G:=Image(IsomorphismPermGroup(SmallGroup(690,1)));;
#! gap> H:=Image(IsomorphismPcGroup(SmallGroup(690,1)));;
#! gap> IsIsomorphicSOTGroups(G,H);
#! true
#! @EndExampleSession
DeclareGlobalFunction("IsIsomorphicSOTGroups");
#! @Description
#! returns <K>true</K> if the order <A>n</A> is available in the &SOTGrps; library, and <K>false</K> otherwise.
#! @Arguments n
DeclareGlobalFunction("IsSOTAvailable");
#! @Description
#! prints information on the groups of the specified order.
#! Since there are some overlaps between the existing SmallGrps library and the &SOTGrps; library.
#! In particular, &SOTGrps; may construct the groups in a different order and so generate a different group ID; we denote such IDs by <K>SOT</K>.
#! If the order covered in &SOTGrps; library has no conflicts with the existing library, then such a flag is removed.
#! @BeginExampleSession
#! gap> SOTGroupsInformation(2^2*3*19);
#!
#! There are 15 groups of order 228.
#!
#! The groups of order p^2qr are either solvable or isomorphic to Alt(5).
#! The solvable groups are sorted by their Fitting subgroup.
#! SOT 1 - 2 are the nilpotent groups.
#! SOT 3 has Fitting subgroup of order 57.
#! SOT 4 - 7 have Fitting subgroup of order 76.
#! SOT 8 - 9 have Fitting subgroup of order 38.
#! SOT 10 - 15 have Fitting subgroup of order 114.
#!
#! gap> SOTGroupsInformation(2662);
#!
#! There are 15 groups of order 2662.
#!
#! The groups of order p^3q are solvable by Burnside's pq-Theorem.
#! These groups are sorted by their Sylow subgroups.
#! 1 - 3 are abelian.
#! 4 - 5 are nonabelian nilpotent and have a normal Sylow 11-subgroup and a
#! normal Sylow 2-subgroup.
#! 6 is non-nilpotent and has a normal Sylow 2-subgroup with Sylow
#! 11-subgroup [ 1331, 1 ].
#! 7 - 9 are non-nilpotent and have a normal Sylow 2-subgroup with Sylow
#! 11-subgroup [ 1331, 2 ].
#! 10 - 12 are non-nilpotent and have a normal Sylow 2-subgroup with Sylow
#! 11-subgroup [ 1331, 5 ].
#! 13 - 14 are non-nilpotent and have a normal Sylow 2-subgroup with Sylow
#! 11-subgroup [ 1331, 3 ].
#! 15 is non-nilpotent and has a normal Sylow 2-subgroup with Sylow
#! 11-subgroup [ 1331, 4 ].
#! @EndExampleSession
#! @Arguments n
DeclareGlobalFunction("SOTGroupsInformation");
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