*(English)*Zbl 1112.20018

Let $G$ be a finite group. If $k$ is a positive integer and $N$ is a normal subgroup of $G$, then a function $\varphi $ from the Cartesian power ${G}^{k}$ to $G$ is said to be compatible with $N$ if $x=({x}_{1},\cdots ,{x}_{k}),y=({y}_{1},\cdots ,{y}_{k})\in {G}^{k}$ and ${x}_{i}{y}_{i}^{-1}\in N$ for $1\le i\le k$ implies $\varphi \left(x\right)\varphi {\left(y\right)}^{-1}\in N$. A function $\varphi :{G}^{k}\to G$ that is compatible with all normal subgroups of $G$ is said to be compatible. A group $G$ is called $k$-affine complete if every compatible function $\varphi :{G}^{k}\to G$ is a polynomial function as defined by *W. Nöbauer* and the reviewer [Algebra of polynomials. North-Holland Mathematical Library. Vol. 5. (1973; Zbl 0283.12101)], and $G$ is called affine complete if $G$ is $k$-affine complete for every positive integer $k$.

Let $A$ be a finite Abelian group. Then $\text{Dih}\left(A\right)$ is defined as the semidirect product of $A$ with the group of order 2 where the conjugation action of its non-identity element on $A$ is inversion. Theorem 1.1: $\text{Dih}\left(A\right)$ is 1-affine complete if and only $A$ is 1-affine complete. Theorem 1.2: If $P$ is the Sylow 2-subgroup of $A$ and $Q$ its complement, then the following conditions are equivalent: (i) $\text{Dih}\left(A\right)$ is affine complete, (ii) $\text{Dih}\left(A\right)$ is 2-affine complete, (iii) $P$ is elementary-Abelian and $Q$ is affine complete.

##### MSC:

20D40 | Products of subgroups of finite groups |

08A40 | Operations on general algebraic systems |

20D60 | Arithmetic and combinatorial problems on finite groups |

16Y30 | Near-rings |

20F05 | Generators, relations, and presentations of groups |