In this section we define skew braces and list some of their main properties [GV17].
A skew brace is a triple \((A,+,\circ)\), where \((A,+)\) and \((A,\circ)\) are two (not necessarily abelian) groups such that the compatibility \(a\circ (b+c)=a\circ b-a+a\circ c\) holds for all \(a,b,c\in A\). Ones proves that the map \(\lambda\colon (A,\circ)\to\mathrm{Aut}(A,+)\), \(a\mapsto\lambda_a(b)\), \(\lambda_a(b)=-a+a\circ b\), is a group homomorphism. Notation: For \(a,b\in A\), we write \(a*b=\lambda_a(b)-b\).
‣ IsSkewbrace( arg ) | ( filter ) |
Returns: true or false
‣ Skewbrace( list ) | ( operation ) |
Returns: a skew brace
The argument list is a list of pairs of elements in a group. By Proposition 5.11 of [GV17], skew braces over an abelian group \(A\) are equivalent to pairs \((G,\pi)\), where \(G\) is a group and \(\pi\colon G\to A\) is a bijective \(1\)-cocycle, a finite skew brace can be constructed from the set \(\{(a_j,g_j):1\leq j\leq n\}\), where \(G=\{g_1,\dots,g_n\}\) and \(A=\{a_1,\dots,a_n\}\) are permutation groups. This function is used to construct skew braces.
gap> Skewbrace([[(),()]]); <brace of size 1> gap> Skewbrace([[(),()],[(1,2),(1,2)]]); <brace of size 2>
‣ SmallSkewbrace( n, k ) | ( operation ) |
Returns: a skew brace
The function returns the k-th skew brace from the database of skew braces of order n.
gap> SmallSkewbrace(8,3); <brace of size 8>
‣ TrivialBrace( abelian_group ) | ( operation ) |
Returns: a brace
This function returns the trivial brace over the abelian group abelian_group. Here abelian_group should be an abelian group!
gap> TrivialBrace(CyclicGroup(IsPermGroup, 5)); <brace of size 5>
‣ TrivialSkewbrace( group ) | ( operation ) |
Returns: a skew brace
This function returns the trivial skew brace over group.
gap> TrivialSkewbrace(DihedralGroup(10)); <skew brace of size 10>
‣ SmallBrace( n, k ) | ( operation ) |
Returns: a brace of abelian type
The function returns the k-th brace (of abelian type) from the database of braces of order n.
gap> SmallBrace(8,3); <brace of size 8>
‣ IdSkewbrace( obj ) | ( attribute ) |
Returns: a list
The function returns [ n, k ] if the skew brace obj is isomorphic to SmallSkewbrace(n,k).
gap> IdSkewbrace(SmallSkewbrace(8,5)); [ 8, 5 ]
‣ AutomorphismGroup( obj ) | ( attribute ) |
Returns: a list
The function computes the automorphism group of a skew brace.
gap> br := SmallSkewbrace(8,20);; gap> AutomorphismGroup(br); <group with 8 generators> gap> StructureDescription(last); "D8"
gap> br := SmallSkewbrace(8,25);; gap> aut := AutomorphismGroup(br);; gap> f := Random(aut);; gap> x := Random(br);; gap> ImageElm(f, x) in br; true
‣ IdBrace( obj ) | ( attribute ) |
Returns: a list
The function returns [ n, k ] if the brace of abelian type obj is isomorphic to SmallBrace(n,k).
gap> IdBrace(SmallBrace(8,5)); [ 8, 5 ]
‣ IsomorphismSkewbraces( obj1, obj2 ) | ( function ) |
Returns: an isomorphism of skew braces if obj1 and obj2 are isomorphic and fail otherwise.
If \(A\) and \(B\) are skew braces, a skew brace homomorphism is a map \(f\colon A\to B\) such that
\[f(a+b)=f(a)+f(b)\quad f(a\circ b)=f(a)\circ f(b)\]
hold for all \(a,b\in A\). A skew brace isomorphism is a bijective skew brace homomorphism. IsomorphismSkewbraces first computes all injective homomorphisms from \((A,+)\) to \((B,+)\) and then tries to find one \(f\) such that \(f(a\circ b)=f(a)\circ f(b)\) for all \(a,b\in A\).
‣ DirectProductSkewbraces( obj1, obj2 ) | ( operation ) |
Returns: the direct product of obj1 and obj2
gap> br1 := SmallBrace(8,18);; gap> br2 := SmallBrace(12,2);; gap> br := DirectProductSkewbraces(br1,br2);; gap> IsLeftNilpotent(br); false gap> IsRightNilpotent(br); false gap> IsSolvable(br); true
‣ DirectProductOp( arg1, arg2 ) | ( operation ) |
‣ IsTwoSided( obj ) | ( property ) |
Returns: true if the skew brace is two sided, false otherwise
A skew brace \(A\) is said to be two-sided if \((a+b)\circ c=a\circ c-c+b\circ c\) holds for all \(a,b,c\in A\).
gap> IsTwoSided(SmallSkewbrace(8,2)); false gap> IsTwoSided(SmallSkewbrace(8,4)); true
‣ IsAutomorphismGroupOfSkewbrace( obj ) | ( property ) |
Returns: true if the group is the automorphism group of a skew braces, false otherwise
gap> br := SmallSkewbrace(8,25);; gap> aut := AutomorphismGroup(br);; gap> Order(aut); 4 gap> IsAutomorphismGroupOfSkewbrace(aut); true
‣ IsClassical( obj ) | ( property ) |
Returns: true if the skew brace is of abelian type, false otherwise
Let \(\mathcal{X}\) be a property of groups. A skew brace \(A\) is said to be of \(\mathcal{X}\)-type if its additive group belongs to \(\mathcal{X}\). In particular, skew braces of abelian type are those skew braces with abelian additive group. Such skew braces were introduced by Rump in [Rum07].
‣ IsOfAbelianType( arg ) | ( property ) |
Returns: true or false
‣ IsBiSkewbrace( obj ) | ( property ) |
Returns: true if the skew brace is a bi-skew brace, false otherwise
A skew brace \((A,+,\circ)\) is said to be a bi-skew brace if \((A,\circ,+)\) is a skew brace
gap> Number([1..NrSmallSkewbraces(8)], k->IsBiSkewbrace(SmallSkewbrace(8,k))); 39
‣ IsOfNilpotentType( obj ) | ( property ) |
Returns: true if the skew brace is of nilpotent type, false otherwise
Let \(\mathcal{X}\) be a property of groups. A skew brace \(A\) is said to be of \(\mathcal{X}\)-type if its additive group belongs to \(\mathcal{X}\). In particular, skew braces of nilpotent type are those skew braces with nilpotent additive group.
‣ IsTrivialSkewbrace( obj ) | ( property ) |
Returns: true if the skew brace is trivial, false otherwise
The function returns true if the skew brace \(A\) is trivial, i.e., \(a\circ b=a+b\) for all \(a,b\in A\). WARNING: The property IsTrivial applied to a skew brace will return true if and only if the skew brace has only one element.
gap> br := SmallSkewbrace(9,1);; gap> IsTrivialSkewbrace(br); true gap> IsTrivial(br); false
‣ Skewbrace2YB( obj ) | ( attribute ) |
Returns: the set-theoretic solution associated with the skew brace obj
If \(A\) is a skew brace, the map \(r_A\colon A\times A\to A\times A\)
\[r_A(a,b)=(\lambda_a(b),\lambda_a(b)'\circ a\circ b)\]
is a non-degenerate set-theoretic solution of the Yang--Baxter equation. Furthermore, \(r_A\) is involutive if and only if \(A\) is of abelian type (i.e., the additive group of \(A\) is abelian).
gap> Skewbrace2YB(TrivialBrace(CyclicGroup(6))); <A set-theoretical solution of size 6>
‣ Brace2YB( arg ) | ( attribute ) |
‣ SkewbraceSubset2YB( obj ) | ( operation ) |
Returns: the set-theoretic solution associated with a given subset of a skew brace
gap> br := TrivialSkewbrace(SymmetricGroup(3));; gap> AsList(br); [ <()>, <(2,3)>, <(1,2)>, <(1,2,3)>, <(1,3,2)>, <(1,3)> ] gap> SkewbraceSubset2YB(br, last{[4,5]}); <A set-theoretical solution of size 2>
‣ SemidirectProduct( A, B, s ) | ( operation ) |
Returns: the semidirect product of skew braces
Let \(A\) and \(B\) be two skew braces and \(\sigma\) be a skew brace action of \(B\) on \(A\), this is a group homomorphism \(\sigma\colon (B,\circ)\to Aut_{\mathrm{Br}}(A)\) from the multiplicative group of \(B\) to the skew brace automorphism of \(A\). The semidirect product of \(A\) and \(B\) with with respect to \(\sigma\) is the skew brace \(A\rtimes_{\sigma}B\) with operations
\[ (a_1,b_1)+(a_2,b_2)=(a_1+a_2,b_1+b_2), \quad (a_1,b_1)\circ (b_2,b_2)=(a_1\circ\sigma(b_1)(a_2),b_1\circ b_2) \]
gap> A := SmallSkewbrace(4,2);; gap> B := SmallSkewbrace(3,1);; gap> s := SkewbraceActions(B,A);; gap> Size(s); 1 gap> IdSkewbrace(SemidirectProduct(A,B,s[1])); [ 12, 11 ] gap> IdSkewbrace(DirectProduct(A,B)); [ 12, 11 ]
‣ UnderlyingAdditiveGroup( A ) | ( attribute ) |
Returns: the underlying multiplicative group of the skew brace
gap> br := SmallBrace(4,2);; gap> G:=UnderlyingMultiplicativeGroup(br);; gap> StructureDescription(G); "C2 x C2"
‣ UnderlyingMultiplicativeGroup( A ) | ( attribute ) |
Returns: the underlying additive group of the skew brace
gap> br := SmallSkewbrace(6,2);; gap> G:=UnderlyingAdditiveGroup(br);; gap> IsAbelian(G); false
generated by GAPDoc2HTML