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Groups of even order are of course groups with an even number of elements. One of their properties is that they always have at least one element of order 2. This can of course be proved by using Sylow's theorem, but that's a bit like using a flamethrower on a mosquito.

Consider a group G of even order. What we do is group all the elements by pairs. For all elements of the group whose order is strictly greater than 2, x and x-1 are distinct. So we have lots of sets {x,x-1} of 2 elements, and also one set containing just the identity. If we have elements of order 2, then x-1=x, so we put them in a set by themselves. if G has no elements of order 2, then you'll notice that leaves us with 2n+1 elements, which is bad, hence we must have at least one element of order 2.

Incidentally, if g is the only element of order 2, then it commutes with any element of G. To proove this, take x in G and consider y=x-1gx. Then y2=g2=e
hence y has order 2. But g is the only element of order 2, therefore y=g
x-1gx=g, therefore gx=xg

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