Is x a Group? Testing the 4 Axioms and Multiplication Table Method

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Hey guys, I'm having an issue with a question, namely

Let x be a subset of S4. Is x a group?

x = {e, (123), (132), (12)(34)}

I don't really understand how I can test the 4 axioms of a group and how x being a subset of S4 would help?
 
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You have only four elements. Why not just write out the entire multiplication table and see if you have a group on your hands?
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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