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Math Help - Lagrange Theorem.

  1. #1
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    Lagrange Theorem.

    Suppose that G is an Abelian group with an odd number of elements. show that the product of all the elements of G is the identity.

    Let |G|=8. Show that G must have and element of order 2.
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    Quote Originally Posted by Juancd08 View Post
    Suppose that G is an Abelian group with an odd number of elements. show that the product of all the elements of G is the identity.
    Hint: Pair x and y so that xy=e. Note why you need the fact that |G| is odd.

    Let |G|=8. Show that G must have and element of order 2.
    Let a\in G - \{e \} and construct \left< a\right>. By Lagrange's theorem |\left< a \right>| = 2,4,8. Since this is a cyclic group and 2 divides its order we can find an element of that order.
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