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2

Part of 1993 Hungary-Israel Binational

Problems(2)

Determine all polynomials such that f(x^2-2x)=f^2(x-2)

Source: 4-th Hungary-Israel Binational Mathematical Competition 1993

5/26/2007
Determine all polynomials f(x)f (x) with real coeffcients that satisfy f(x2−2x)=f2(x−2)f (x^{2}-2x) = f^{2}(x-2) for all x.x.
algebrapolynomialalgebra proposed
group with x\to x^n is an isomorphism

Source: 4-th Hungary-Israel Binational Mathematical Competition 1993

5/26/2007
In the questions below: GG is a finite group; H≤GH \leq G a subgroup of G;∣G:H∣G; |G : H | the index of HH in G;∣X∣G; |X | the number of elements of X⊆G;Z(G)X \subseteq G; Z (G) the center of G;G′G; G' the commutator subgroup of G;NG(H)G; N_{G}(H ) the normalizer of HH in G;CG(H)G; C_{G}(H ) the centralizer of HH in GG; and SnS_{n} the nn-th symmetric group. Suppose that n≥1n \geq 1 is such that the mapping x↦xnx \mapsto x^{n} from GG to itself is an isomorphism. Prove that for each a∈G,an−1∈Z(G).a \in G, a^{n-1}\in Z (G).
group theoryabstract algebrasuperior algebrasuperior algebra unsolved