For X ≠ ∅, an action of G on X is transitive if and only if, given x and y in X, there is some g ∈ G such that y = gx.

Expert Solution

Proof: Suppose the action is transitive, so there is one orbit. Given x in X, its orbit must

fill up X, so every element of X has the form gx for some g ∈ G.

Conversely, suppose that for each x and y in X we can write y = gx for some g ∈ G. Fix

x ∈ X. Since every y ∈ X has the form gx for some g, every y is in the orbit of x. Thus X has only one orbit.

Suppose the action is transitive, so there is one orbit. Given x in X, its orbit must

fill up X, so every element of X has the form gx for some g ∈ G.

Conversely,

suppose that for each x and y in X we can write y = gx for some g ∈ G. Fix x ∈ X. Since every y ∈ X has the form gx for some g, every y is in the orbit of x. Thus X has only one orbit.

Nipun Maity answered 2 years ago

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