show even and odd

**mandy123** · September 17th 2008, 04:29 PM

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

**ThePerfectHacker** · September 17th 2008, 06:00 PM

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

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