# Thread: show even and odd

1. ## show even and odd

Let G be a finite group
a) Show that the number of elements x of G such that x^3=e is odd
b) Show that the number of elements x of G such that x^2 does not equal e is even

2. Originally Posted by mandy123
Let G be a finite group
a) Show that the number of elements x of G such that x^3=e is odd
Let $X = \{ x\in G | x^3 = e\}$. Now if $x\not = e$ is in $X$ then $x^3 = e$ so $x$ has order $3$. Therefore, $x\not = x^2$. But $(x^2)^3 = (x^3)^2 = e$. Thus, $x^2 \in X$. Let $X = \{ e,x_1,x_2,....,x_n,x_{n+1} \}$. Write them in such a way so that $x_1 = x_2^2,....,x_n=x_{n+1}^2$. Therefore, all $x_i$ can be paired with $x_{i+1}$. Thus, there are an even number (zero is allowed) of elements which have order three. But we also include $e$ and this makes the total number of elements odd.

b) Show that the number of elements x of G such that x^2 does not equal e is even
Similar to above.