Any pentagonal quasigroup Q Q is proved to have the product x y = φ ( x ) + y − φ ( y ) xy=\varphi \left(x)+y-\varphi (y) , where ( Q , + ) \left(Q,+) is an Abelian group, φ \varphi is its regular automorphism satisfying φ 4 − φ 3 + φ 2 − φ + ε = 0 {\varphi }^{4}-{\varphi }^{3}+{\varphi }^{2}-\varphi +\varepsilon =0 and ε \varepsilon is the identity mapping. All Abelian groups of order n < 100 n\lt 100 inducing pentagonal quasigroups are determined. The variety of commutative, idempotent, medial groupoids satisfying the pentagonal identity ( x y ⋅ x ) y ⋅ x = y \left(xy\cdot x)y\cdot x=y is proved to be the variety of commutative, pentagonal quasigroups, whose spectrum is { 1 1 n : n = 0 , 1 , 2 , … } \left\{1{1}^{n}:n=0,1,2,\ldots \right\} . We prove that the only translatable commutative pentagonal quasigroup is x y = ( 6 x + 6 y ) ( mod 11 ) xy=\left(6x+6y)\left({\rm{mod}}\hspace{0.33em}11) . The parastrophes of a pentagonal quasigroup are classified according to well-known types of idempotent translatable quasigroups. The translatability of a pentagonal quasigroup induced by the group Z n {{\mathbb{Z}}}_{n} and its automorphism φ ( x ) = a x \varphi \left(x)=ax is proved to determine the value of a a and the range of values of n n .

中文翻译：

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五角准群，可翻译性和灾难性

证明任何五边形准群QQ的乘积为xy =φ（x）+ y-φ（y）xy = \ varphi \ left（x）+ y- \ varphi（y），其中（Q，+）\ left（ Q，+）是一个阿贝尔群，φ\ varphi是满足φ4 −φ3 +φ2 −φ+ε= 0 {\ varphi} ^ {4}-{\ varphi} ^ {3} + { \ varphi} ^ {2}-\ varphi + \ varepsilon = 0，并且ε\ varepsilon是身份映射。确定所有n <100 n \ lt 100阶诱导五边形拟群的阿贝尔群。满足五边形（xy⋅x）y⋅x = y \ left（xy \ cdot x）y \ cdot x = y的交换，幂等，中间类群的种类被证明是交换，五边形准群的种类频谱为{1 1 n：n = 0，1，2，…} \ left \ {1 {1} ^ {n}：n = 0,1,2，\ ldots \ right \}。我们证明唯一可翻译的交换五边形准群是xy =（6 x + 6 y）（mod 11）xy = \ left（6x + 6y）\ left（{\ rm {mod}} \ hspace {0.33em} 11） 。五边形准群的突变是根据幂等可翻译准群的众所周知的类型进行分类的。Z n {{\\ mathbb {Z}}} _ {n}诱导的五边形准群的可翻译性及其自同构φ（x）= ax \ varphi \ left（x）= ax决定了aa和nn的值范围。