Preliminaries

In this section we define skew braces and list some of their main properties .
Definition and examples 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. true or false

a skew brace The argument list is a list of pairs of elements in a group. By Proposition 5.11 of , 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>
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>
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>
a skew brace This function returns the trivial skew brace over group.
gap> TrivialSkewbrace(DihedralGroup(10)); <skew brace of size 10>
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>
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 ]
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 ]
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. 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
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
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 . 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. 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>

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>