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Interesting abelian subgroup of the complex numbers

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December 13, 2019
functiongroup theoryabstract algebracomplex numbers

Problem Statement

Let be the set G={(u,v)∈C2∣u≠0} G=\{ (u,v)\in \mathbb{C}^2| u\neq 0 \} and a function φ:C∖{0}⟶C∖{0} \varphi :\mathbb{C}\setminus\{ 0\}\longrightarrow\mathbb{C}\setminus\{ 0\} having the property that the operation ∗:G2⟶G *:G^2\longrightarrow G defined as (a,b)∗(c,d)=(ac,bc+dφ(a)) (a,b)*(c,d)=(ac,bc+d\varphi (a)) is associative.
a) Show that (G,∗) (G,*) is a group. b) Describe φ, \varphi , knowing that (G,∗)(G,*) is a commutative group.
Marius Perianu
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