Construction of Finite Braces

Classification of the affine structures of a generalized quaternion group of order ⩾32{\geqslant 32}

“…In the extreme case ha 4 i D Soc.A/, the retraction A=Soc.A/ of A is a dihedral brace of order 8. These braces were classified by Bachiller [2] and further investigated in [16]. There are 8 such braces, but we show that only one of them can arise.…”

“…Now there are eight braces with adjoint group D 4 (see [2]). In [16,Example 3], they are denoted as B 1 ; : : : ; B 8 . For B 1 ; : : : ; B 6 , the socle is non-trivial, which implies that ha 2 i is an ideal.…”

“…Our first aim is to determine the possible types of braces B D A=Soc.A/. In[16, Example 3], the dihedral braces B 1 ; : : : ; B 8 of order 8 are described. The cyclic brace B 1 is excluded by the following.…”

One-generator braces and indecomposable set-theoretic solutions to the Yang–Baxter equation

“…Thus, an involutive q-brace is a cycle set, but not necessarily a brace. Indeed, [36,Theorem 1] shows that a brace is equivalent to an involutive q-brace satisfying…”

“…Example 3. The symmetric group S 4 is the adjoint group of three braces (see [36,Example 4]). For two of them, the 2-component is irretractable, and only these two braces A with A • ∼ = S 4 admit a transitive cycle basis.…”