Sarah: algebra///////////////////////////////////////////////////////?
let G be a group of order 2n. Prove that G has an element of order 2..
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Answer by ANDREW K
Group G order 2n means sub groups at least 2 and n…and any mixtures of 2 and n
therefore there is an element of order 2
Let S be the set of elements in G that have order greater than two. If x is in S, then x cannot equal its inverse, or else x^2 = e. This means we can pair each element of S with it’s inverse, and so S has an even number of elements. Then SU{e} has an odd number of elements (since e is not in S). Since G has an even number of elements, there must be some g in G that is not in SU{e}. By definition of S, this element g has order less than three. Furthermore g is not the identity (so does not have order 1). Therefore g has order two.
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