The group is abelian if and only if this table is symmetric about the main diagonal. This is true since the group is abelian iff for all , which is iff the entry of the table equals the entry for all , i.e. the table is symmetric about the main diagonal.
The group is abelian if and only if this table is symmetric about the main diagonal. This is true since the group is abelian iff for all  which is iff the entry of the table equals the entry for all , i.e. the table is symmetric about the main diagonal