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Source: Romania District Olympiad 2013,grade XII(problem 2)

March 14, 2013
superior algebrasuperior algebra unsolved

Problem Statement

Problem 2. A group (G,⋅)\left( G,\cdot \right) has the propriety(P)\left( P \right), if, for any automorphism f for G,there are two automorphisms g and h in G, so that f(x)=g(x)⋅h(x)f\left( x \right)=g\left( x \right)\cdot h\left( x \right), whatever x∈Gx\in Gwould be. Prove that: (a) Every group which the property (P)\left( P \right) is comutative. (b) Every commutative finite group of odd order doesn’t have the (P)\left( P \right) property. (c) No finite group of order 4n+2,n∈N4n+2,n\in \mathbb{N}, doesn’t have the (P)\left( P \right)property. (The order of a finite group is the number of elements of that group).
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