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Given a group G and a space Y , then an action of G on Y is a homomorphism ρ from G to the group Homeo(Y ) of all homeomorphisms from Y to itself. Thus to each g ∈ G is associated a homeomorphism ρ(g) : Y →Y.

now the following problem:

Letφ:R2 →R2 be the linear transformation φ(x, y) = (2x, y/2). This generates an action of Z on X = R2−{0}. Show this action is a covering space action and compute π1(X/Z).

I don't understand the question, going by the definition Z is G and each integer is associated with a homeomorphism from R2−{0} to R2−{0}. So how exactly does the linear transformation generate the action. Is φ the homeomorphism (kx,y/k) associated with k in Z (in this case 2) ?