Question

Consider the group (Rˣ,⋅) of non-zero real numbers under multiplication.

(a) Find all r ∈ Rˣ such that ⟨r⟩ is a finite group. Be sure to justify your answer!

Answer 1

The values of r ∈ Rˣ such that ⟨r⟩ is a finite group are r = 1 and r = -1.

Step-by-step explanation:

To find all r ∈ Rˣ such that ⟨r⟩ is a finite group, we need to determine which values of r satisfy the condition that every power of r eventually equals 1.

In this case, r is a non-zero real number, so we know that r ≠ 0. The only way for every power of r to equal 1 is if r is either 1 or -1.

Therefore, the values of r ∈ Rˣ such that ⟨r⟩ is a finite group are r = 1 and r = -1.