Concept guide
Finite abelian group: from a complete Cayley table, group qualification first, then commutativity
On the carrier G={0,1,2,3} a complete 4×4 Cayley table is given (the cell in row a, column b is a*b): row 0 is (1,2,3,0), row 1 is (2,3,0,1), row 2 is (3,0,1,2), row 3 is (0,1,2,3). Check closure, associativity, a unique identity and inverses in that order, then check a*b=b*a for all a, b, to decide whether (G,*) is a finite abelian group; find the orders of the elements 0 and 1 and the cyclic subgroups they generate; and use the symmetric table a∘b=|a−b| on {0,1,2} to show that a symmetric table cannot replace group qualification.