Assuming that G is a finite cyclic group, let "a" be the product of all the elements in the group.
i. If G has odd order, then a=e. Is this because there are an even number of non-trivial elements must have their inverses within the non-trivial factors within the product?
ii. If G has even order then a is not equal to e. Here there an odd number of non-trivial elements. There must also be an odd number of elements who are their own inverses, and therefore a cannot equal e?
Thanks for any help! I'm just reading a textbook and trying practice problems!