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?
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 .