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Affine completeness of generalised dihedral groups. (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 $x=(x_1,\dots,x_k), y=(y_1,\dots,y_k)\in G^k$ and $x_iy_i^{-1}\in N$ for $1\le i\le k$ implies $\varphi(x)\varphi(y)^{-1}\in N$. A function $\varphi\colon G^k\to G$ 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 $\varphi\colon G^k\to G$ is a polynomial function as defined by {\it 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 $\text{Dih}(A)$ 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: $\text{Dih}(A)$ 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) $\text{Dih}(A)$ is affine complete, (ii) $\text{Dih}(A)$ 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
##### Keywords:
affine complete groups
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