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hgj

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**Group Theory, please help!**

Okay, so I'm stuck on a couple questions from my homework, and any guidance would be much appreciated.

1. Prove that if G is a finite group with an even number of elements,

then there is an element x in G such that x is not the identity and

x^2 = e.

I know there exists some element x in G because G is not empty. And because e (the identity element in G) is unique, x is not equal to e, so x is not the identity. But I can't see how to go from x doesn't equal e to x^2 = e.

2. Prove that if (S,*) is a finite set with a binary operation that is

associative, has an identity, and satisfies the cancellation laws,

then (S,*) is a group.

I know that for (S, *) to be a group, it must be associative, there must exist an identity element e in S wrt *, and every element in S must be invertible. The first two properties follow easily from the way (S,*) is defined, but I don't know how to show the last property holds. It makes sense to me that it's true when I look at the cancellation law (if a,b,c are in S and ab = ac, the b = c), and I've tried working backwards, but then I find myself wanting to create an inverse in S, and that seems wrong.

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