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This time around I need to prove that a finite group of order 10 must contain an element of order 2 and an element of order 5.

If the group is cyclic then this is trivial. So assuming the group is not cyclic, it's easy to show that there exists an element of order 2 in the group. And it is just as trivial to show that any element, except for e, must be of either order 2 or order 5. But I'm having a hard time proving that theremustbe an element of order 5 in the group. IthinkI just need to show that it's impossible to have every element in the group be of order 2, only problem is I don't know how... I do know that if a group consists only of elements of order 2, then it must be abelian. So perhaps it would be just as good to show that a non-cyclic group of order 10 cannot be abelian?

I'd much appreciate hints and nudges in the right direction. :)

Chen

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# Finite groups and order of their elements

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