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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 ϕ from the Cartesian power G k to G is said to be compatible with N if x=(x 1 ,,x k ),y=(y 1 ,,y k )G k and x i y i -1 N for 1ik implies ϕ(x)ϕ(y) -1 N. A function ϕ:G k 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 ϕ:G k 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 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: 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) Dih(A) is affine complete, (ii) Dih(A) is 2-affine complete, (iii) P is elementary-Abelian and Q is affine complete.

MSC:
20D40Products of subgroups of finite groups
08A40Operations on general algebraic systems
20D60Arithmetic and combinatorial problems on finite groups
16Y30Near-rings
20F05Generators, relations, and presentations of groups