Mathematical Systems Question Help Alright, this is another question for my math for teachers course. This question is not actually in the homework, but there are problems similar to it. I'd really like to learn how to do problems like this one, so it would be much appreciated if I was given a solid explanation/demonstration. I don't really have a clue where to start, so I've defined the terms as best as I can under each question... The question should be answered using the data from the table.
A mathematical system is defined by the following table:

Is there an identity element in this system? If so, what is it?
Identity is: aXe=a and eXa=a
Is closure satisfied by this system? Explain.
Closure is: aXb
Is this system commutative? Explain.
Commutative is: aXb=bXa
Does every element have an inverse? Explain.
Inverse is: aXx=e and xXa=e
Is this system a group? Explain.
Not sure what they mean by this... 
Are they referring to another property?. 

As we can see through the table below: 


*

*According to the table if we want to have an identity element, we should see one of the following cases:
        a    b    c
     ---------------
   a .  a    b    c
   b .  b
   c .  c

Or:
        a    b    c
     ---------------
   a .       a     
   b .  a    b    c
   c .       c

Or: 
        a    b    c
     ---------------
   a .            a
   b .            b
   c .  a    b    c

so we see the second form above is coincide the table. It is not difficult that $b$ plays a role of an identity here. Just check the definition you had written for elements $a, b$ and $c$.


*

*If $id=b$ as it happened above, so search for an element $x$ among $a,b,c$ such that $$a*x=b,~~x*a=b$$ It is really $x=a$ and $x=c$. This shows that the structure is not a Group since the inverse is unique in any Group and for any element in it. Now do the same way for $b$ and $c$ to find additional information.

