Is it a group?

In abstract alegbra a group is a set and a binary operation. Our sets will contain non-integers and we are just going to deal with finite groups. The binary operation will be a function that takes in two elements of the set and returns a single value.

Your task in this kata is to create a group object and a function that decides if the group is valid.

In order for a set and binary operation to be a group it must satisfy four conditions: Let a,b,c be elements in our set and * our binary operation (function)

1. The set must be closed under the binary operation: a*b is an element of the set.
   (The output from given_function(a,b) is an element of the set)

2. The set must be associative under the binary operation:
   (a*b)*c=a*(b*c)
(given_function(given_function(a,b),c) = given_function(a, given_function(b,c)) )

3. There must be an identity element i so that for any other element of the set a:
   i*a=a=a*i (given_function(i,a)=a=given_function(a,i))

4. For each element a there must be an inverse element a^-1 in the set, such that: a*a^-1=i=a^-1*a (given_function(a,a^-1)=identity)

The wikipedia page is really useful for a clearer/thorougher understanding. I would also reccomend looking up Cayley tables and designing a function to create one for a group object, makes it much easier to see where you have gone wrong.

https://en.wikipedia.org/wiki/Group_(mathematics)#Definition_and_illustration