[loops] 6 Methods based on permutation groups

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# 6 Methods based on permutation groups

### Sections

1. Parent of a quasigroup
2. Comparing quasigroups with common parent
3. Subquasigroups and subloops
4. Translations and sections
5. Multiplication groups
6. Inner mapping groups
7. Nuclei, commutant, center, and associator subloop
8. Normal subloops
9. Factor loops
10. Nilpotency and central series
11. Solvability
12. Isomorphisms and automorphisms
13. How are isomorphisms computed
14. Isotopisms

Most calculations in the LOOPS package are delegated to groups, taking advantage of the various permutations and permutation groups associated with quasigroups. This chapter explains in detail how the permutations associated with a quasigroup are calculated, and it also describes some of the core methods of LOOPS based on permutations. Additional core methods can be found in Chapter Testing properties of quasigroups and loops.

## 6.1 Parent of a quasigroup

Let Q be a quasigroup and S a subquasigroup of Q. Since the multiplication in S coincides with the multiplication in Q, it is reasonable not to store the multiplication table of S. However, the quasigroup S then must know that it is a subquasigroup of Q. In order to facilitate this relationship, we introduce the attribute

- `Parent(` Q`) A`

  for a quasigroup Q.

  When Q is not created as a subquasigroup of another quasigroup, the attribute `Parent(` Q`)` is set to Q. When Q is created as a subquasigroup of a quasigroup H, we let `Parent(` Q`) := Parent(` H`)`. Thus, in effect, `Parent(` Q`)` is the largest quasigroup from which Q has been created.

  Let Q be a quasigroup with parent P, where P is some n-element quasigroup. Let x be an element of Q. Then `x![1]` is the position of x among the elements of P, i.e., `x![1] = Position( Elements( P ), x )`. The position of x among the elements of Q is obtained via

- `Position(` Q`, x ) O`

  While referring to elements of Q by their positions, we therefore must decide if the positions are meant among the elements of Q, or among the elements of P. Since it requires no calculation to obtain `x![1]`, we always use the position of an element in its parent quasigroup. In this way, many attributes of a quasigroup, including its Cayley table, are permanently tied to its parent. It is now clear why we have not insisted that Cayley tables of quasigroups must have entries covering the entire interval 1, ..., m, for some m.

  When S is a list of quasigroup elements, not necessarily from the same quasigroup, the operation

- `PosInParent(` S`)`

  returns the list of positions of elements of S in the corresponding parent, i.e., `PosInParent(` S`)[ i ] = S[ i ]![ 1 ] = Position( Parent( S[ i ] ), S[ i ] )`.

  ## 6.2 Comparing quasigroups with common parent

  noindentAssume that A, B are two quasigroups with common parent Q. Let GA, GB be the canonical generating sets of A and B, respectively, obtained by the method `GeneratorsSmallest`, described above. Then we define A < B if and only if GA < GB lexicographically.

  Note that if A is a subquasigroup of B, we get A < B, but not necessarily vice versa.

  ## 6.3 Subquasigroups and subloops

  When S is a subset of a quasigroup Q (loop L), the smallest subquasigroup of Q (subloop of L) generated by S is returned via:

- `Subquasigroup(` Q`,` S`) O`
- `Subloop(` L`,` S`) O`

  In fact, we allow S to be a list of integers, too, representing the positions of the respective elements in the parent quasigroup (loop).

  If S is empty, `Subloop(` L`,` S`)` returns the one-element subloop of L, while `Subquasigroup(` Q`,` S`)` returns the empty set. The empty set is obviously not a subquasigroup of Q, but this convention is useful for handling certain situation, for instance when the user calls `Center(` Q`)` for a quasigroup Q with empty center.

  The following two operations test if a quasigroup (loop) S is a subquasigroup (subloop) of a quasigroup Q. They return `true` if and only if Q and S have the same parent, and if S is a subset of Q.

- `IsSubquasigroup(` Q`,` S`) O`
- `IsSubloop(` Q`,` S`) O`

  The operation

- `AllSubloops(` L`) O`

  returns a list of all subloops of the loop L.

  Let S be a subloop of L. The list of all right cosets of S in L is obtained via

- `RightCosets(` L`,` S`) F`

  The coset S is listed first, and the elements of each coset are ordered in the same way as elements of S, i.e., if S = [s1,...,sm] then Sx=[s1x, ..., smx].

  When S is a subloop of L, the right transversal of S with respect to L is a subset of L containing one element from each right coset of S in L. It is obtained by

- `RightTransversal(` L`,` S`) O`

  and it returns the first element from each right coset obtained by `RightCosets(` L`,` S`)`.

  ## 6.4 Translations and sections

  When x is an element of a quasigroup Q, the left translation Lx is a permutation of Q. In LOOPS, all permutations associated with quasigroups and their elements are permutations in the sense of GAP, i.e., bijections of some interval 1, ..., n. Moreover, following our convention, the numerical entries of the permutation point to the positions among elements of `Parent(` Q`)`, not Q.

  The left and right translations by x in Q are obtained by

- `LeftTranslation(` Q`,` x`) O`
- `RightTranslation(` Q`,` x`) O`

  The following two attributes calculate the left and right section of a quasigroup Q:

- `LeftSection(` Q`) A`
- `RightSection(` Q`) A`

  Here is an example illustrating the main features of the subquasigroup construction and the relationship between a quasigroup and its parent.

  Note how the Cayley table of the subquasigroup is created only upon explicit demand. Also note that changing the names of elements of a subquasigroup (subloop) automatically changes the names of the elements of the parent subquasigroup (subloop). This is because the elements are shared.

      gap> M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.5 ] );
      <loop of order 3>
      gap> [ Parent( S ) = M, Elements( S ), PosInParent( S ) ];
      [ true, [ l1, l3, l5], [ 1, 3, 5 ] ]
      gap> HasCayleyTable( S );
      false
      gap> SetLoopElmName( S, "s" );; Elements( S ); Elements( M );
      [ s1, s3, s5 ]
      [ s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12 ]
      gap> CayleyTable( S );
      [ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
      gap> LeftSection( S );
      [ (), (1,3,5), (1,5,3) ]
      gap> [ HasCayleyTable( S ), Parent( S ) = M ];
      [ true, true ]
      gap> L := LoopByCayleyTable( CayleyTable( S ) );; Elements( L );
      [ l1, l2, l3 ]
      gap> [ Parent( L ) = L, IsSubloop( M, S ), IsSubloop( M, L ) ];
      [ true, true, false ]
      gap> LeftSection( L );
      [ (), (1,2,3), (1,3,2) ]

  ## 6.5 Multiplication groups

  The left multiplication group, right multiplication group and the multiplication group of a quasigroup Q are calculated as follows:

- `LeftMultiplicationGroup(` Q`) A`
- `RightMultiplicationGroup(` Q`) A`
- `MultiplicationGroup(` Q`) A`

  Let S be a subloop of a loop L. Then the relative left multiplication group of L with respect to S is the group 〈L(x)|x ∈ S〉, where L(x) is the left translation by x in Q restricted to S. The relative right multiplication group and relative multiplication group are defined analogously.

- `RelativeLeftMultiplicationGroup(` L`,` S`) O`
- `RelativeRightMultiplicationGroup(` L`,` S`) O`
- `RelativeMultiplicationGroup(` L`,` S`) O`

  See Section Realizing groups as multiplication groups of loops for more on that topic.

  ## 6.6 Inner mapping groups

  The inner mapping group of a loop L is the stabilizer of the unit element in Mlt(L). The elements of this stabilizer are called inner maps of L.

  The left inner mapping group of a loop L is the stabilizer of the unit element in LMlt(L). The right inner mapping group is defined dually.

  Equivalently, the left inner mapping group is generated by all left inner mappings L(x,y) = Lyx−1LyLx, and the right inner mapping group is generated by all right inner mappings R(x,y) = Rxy−1RyRx.

  In analogy with group theory, we define the conjugation, or the middle inner mapping by x as T(x) = Lx−1Rx. The middle inner mapping grroup is then the subgroup of Mlt(L) generated by all conjugations.

  The corresponding commands in LOOPS are

- `LeftInnerMapping(` L`,` x`,` y`) O`
- `MiddleInnerMapping(` L`,` x`) O`
- `RightInnerMapping(` L`,` x`,` y`) O`
- `LeftInnerMappingGroup(` L`) A`
- `MiddleInnerMappingGroup(` L`) A`
- `RightInnerMappingGroup(` L`) A`
- `InnerMappingGroup(` L`) A`

  Here is an example for multiplication groups and inner mapping groups:

      gap> M := MoufangLoop( 12, 1 );
      <Moufang loop 12/1>
      gap> LeftSection( M )[ 2 ];
      (1,2)(3,4)(5,6)(7,8)(9,12)(10,11)
      gap> Mlt := MultiplicationGroup( M ); Inn := InnerMappingGroup( M );
      <permutation group of size 2592 with 23 generators>
      Group([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ])
      gap> Size( Inn );
      216

  ## 6.7 Nuclei, commutant, center, and associator subloop

  Let Q be a quasigroup. The left nucleus Nλ(Q) of Q is the set {x ∈ Q| x(yz)=(xy)z for every y, z ∈ Q}. One defines similarly the middle nucleus Nμ(Q) and the right nucleus Nρ(Q). Then the nucleus N(Q) of Q is the intersection of the three nuclei.

  The nuclei are calculated in LOOPS as follows:

- `LeftNucleus(` Q`) A`
- `MiddleNucleus(` Q`) A`
- `RightNucleus(` Q`) A`
- `Nuc(` Q`) A`

  Unfortunately, the word `Nucleus` is reserved in the core of GAP for a global function with two variables. That is the reason why we have used the abbreviation `Nuc`, which is also common in the literature. However, we support these synonyms of `Nuc`:

- `NucleusOfLoop(` Q`) A`
- `NucleusOfQuasigroup(` Q`) A`

  Since all nuclei are subquasigroups of Q, they are returned as subquasigroups (resp. subloops). When Q is a loop then all nuclei are in fact groups, and they are returned as associative loops.

  The commutant C(Q) of Q is the set {x ∈ Q | xy=yx for every y ∈ Q}. It is obtained via

- `Commutant(` Q`) A`

  The center Z(Q) is defined as the intersection of C(Q) and N(Q), and it is obtained via

- `Center(` Q`) A`

  It is a subgroup of Q and is therefore returned as an associative loop.

  Finally, the associator subloop of a loop L is the smallest normal subloop A(L) of L containing all associators of L. Equivalently, A(L) is the smallest normal subloop K such that L/K is associative. We use another equivalent reformulation for the purposes of computation: A(L) is the smallest normal subloop of L containing {x\α(x) | x ∈ L, α is a left inner mapping}.

- `AssociatorSubloop(` L`) A`

  ## 6.8 Normal subloops

  A subloop S of a loop L is normal if it is invariant under all inner mappings of L. Normality is tested via:

- `IsNormal(` L`,` S`) O`

  When S is a subset of a loop L or a subloop of L, the normal closure of S in L is the smallest normal subloop of L containing S. It is obtained by

- `NormalClosure(` L`,` S`) O`

  A loop L is simple if all normal subloops of L are trivial. The corresponding test in LOOPS is:

- `IsSimple(` L`) O`

  ## 6.9 Factor loops

  When N is a normal subloop of a loop L, the factor loop L/N can be obtained directly via the command `L/N`, or by

- `FactorLoop(` L`,` N`) O`

  The natural projection from L to L/N is returned by

- `NaturalHomomorphismByNormalSubloop(` L`,` N`) O`

  Here is an illustrating example:

      gap> M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.3 ] );
      <loop of order 3>
      gap> IsNormal( M, S );
      true
      gap> F := FactorLoop( M, S );
      <loop of order 4>
      gap> NaturalHomomorphismByNormalSubloop( M, S );
      MappingByFunction( <loop of order 12>, <loop of order 4>,
          function( x ) ... end )

  ## 6.10 Nilpotency and central series

  The definition of nilpotency and nilpotence class is the same as in group theory. The corresponding commands are:

- `NilpotencyClassOfLoop(` L`) A`
- `IsNilpotent(` L`) P`

  When L is not nilpotent, `NilpotencyClassOfLoop(` L`)` returns `fail`.

  A loop L is said to be strongly nilpotent if its multiplication group is nilpotent. This property is obtained by

- `IsStronglyNilpotent(` L`) P`

  Let L be a loop. Define iterated centers Zi(L) as follows: Z0(L)=Z(L), Zi+1(L) = πi−1( Zi(L) ), where πi is the canonical projection L→ L/Zi(L). The longest series Zi(L), Zi−1(L), ..., Z0(L) with Zi(L) > Zi−1(L) > … > Z0(L) is called the upper central series of L, and is returned via

- `UpperCentralSeries(` L`) A`

  The lower central series, defined in the usual way, is obtained by

- `LowerCentralSeries(` L`) A`

  ## 6.11 Solvability

  The definition of solvability, derived subloop, derived length, Frattini subloop and Frattini factor size is the same as for groups. Frattini subloop is calculated only for strongly nilpotent loops.

- `IsSolvable(` L`) P`
- `DerivedSubloop(` L`) A`
- `DerivedLength(` L`) A`
- `FrattiniSubloop(` L`) A`
- `FrattinifactorSize(` L`) A`

  ## 6.12 Isomorphisms and automorphisms

  All isomorphisms between two loops can be found with LOOPS. The operation

- `IsomorphismLoops(` L`,` M`) O`

  returns a single isomorphism between loops L, M if the loops are isomorphic, and `fail` otherwise.

  If an isomorphism exists, it is returned as a permutation p of 1, ..., |L|, where ip=j means that the ith element of L is mapped onto the jth element of M. This is true even if L or M are not their own parents.

  Since one frequently needs to filter a list of loops up to isomorphism, we support

- `LoopsUpToIsomorphism(` ls`) O`

  Given a list ls of loops, the operation returns a sublist of ls containing one loop from each isomorphism class of loops present in ls.

  The attribute

- `AutomorphismGroup(` L`) A`

  returns the automorphism group of the loop L, with the same convention on permutations as in the case of `IsomorphismLoops`.

  Since two isomorphisms "differ" by an automorphism, all isomorphisms can be obtained by the above two functions.

  While dealing with Cayley tables, it is often useful to rename or reorder the elements without changing the isomorphism type. When Q is a quasigroup of size n and p is a permutation of {1,...,n},

- `IsomorphicCopyByPerm( Q, p )`

  returns quasigroup (Q,°) such that p(xy) = p(x)°p(y), i.e., x°y = p( p−1(x)p−1(y)). When Q is a declared loop, a loop is returned. Consequently, when Q is a declared loop and p(1) = k ≠ 1, then p is first replaced by p°(1,k), to make sure that the Cayley table is normalized.

  When S is a normal subloop of L,

- `IsomorphicCopyByNormalSubloop( L, S )`

  returns an isomorphic copy of L in which the elements are ordered according to the right cosets of S. In particular, the Cayley table of S will appear in the top left corner of the Cayley table of the resulting loop.

  ## 6.13 How are isomorphisms computed

  In order to speed up the search for isomorphisms and automorphisms, we first calculate some loop invariants preserved under isomorphisms, and use these invariants to partition the loop into blocks of elements preserved under isomorphisms. These invariants for a loop L can be obtained via

- `Discriminator(` L`) O`

  Since the details are technical, we will not present them here. See Vo or the file `iso.gi` for more.

  If two loops have different discriminators, they are not isomorphic. If they have identical discriminators, they may or may not be isomorphic. The operation

- `AreEqualDiscriminators(` D1`,` D2`) O`

  returns `true` if the discriminators D1, D2 are equal.

  ## 6.14 Isotopisms

  The package contains a slow method for testing if two loops are isotopic:

- `IsotopismLoops(` K`,` L`) O`

  It returns `fail` if K, L are not isotopic, else it returns an isotopism as a triple of bijections on {1,...,|K|}.

  The method works as follows: It is well known that if a loop K is isotopic to a loop L then there exist a principal loop isotope P of K such that P is isomorphic to L. The algorithm first finds all principal isotopes of K, then filters them up to isomorphism, and then checks if any of them is isomorphic to L. This is rather slow already for small orders, say 30.

  The function

- `LoopsUpToIsotopism(` ls`) O`

  filters the list ls in a way similar to `LoopsUpToIsomorphism`, but using isotopism as the underlying equivalence relation.

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