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Math Help - show even and odd

  1. #1
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    Question 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
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  2. #2
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    Quote Originally Posted by mandy123 View Post
    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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