Let be a group. Then
(a) has a unique identity, and
(b) each element in has a unique inverse.
The table below shows an abelian group with elements:
I can see the identity is show on the first row and column, but I don't see the inverse of each elements. Does it mean that the inverses are not in the group?
A group (G, *) is always associative by definition, but not always commutative. It is commutative if and only if it is abelian, by definition.
In general the inverse of is denoted . Also keep in mind that it's possible to have , so your enumeration of the set above could have duplicates. There would also be duplicates if you listed a pair of inverses twice, for example if .
A quick check to see that this is not a group is to note that in each row and column of the Cayley table every element must occur. This reflects the fact that one can get from each element of a group to every other, for example one can get to from by (post-)multiplying by .