There are two main notational conventions for abelian groups – additive and multiplicative.
See also list of small groups for finite abelian groups of order 30 or less.
One can check that this yields the orders in the previous examples as special cases (see Hillar, C., & Rhea, D.).
The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication). Some important topics in this area of study are: