Question

Show that in a group G of odd order, the equation x²= e has a unique solution.

Further, show that x2= g has a unique solution ∀g∈G,g ≠e .

Answer 1

The equation $x^2=g$ will not have a unique solution if $\exist a,b \in G$ such that $a^2=b^2$ or $\nexists a \in G$ such that $a^2=g.$ So $x^2=a$ has a unique solution if and only if the function $f:G\to G$ such that $f(x)=x^2$ is bijective. Hence by (a), our result follows.