The product of any two transformations of this set belongs to the set. We say that is closed under the operation matrix multiplication. S 3 and are two examples of groups of transformations. G is a subgroup of G if and only if H is also a group. Different types of homomorphisms between groups.
Every group belongs to a group class called an abstract group.
Isomorphic groups are said to have the same structure because they have the same group-theoretic properties. Any given group is isomorph to a given abstract group and its structure is characterized by all the abstract subgroups of this abstract group. Group theory is the study of those properties of groups which are preserved under isomorphism. For finite-order groups this is most clearly done by setting up a group table. Note that by changing the symbols denoting the group elements, or changing the order of the group elements a group table can be made to look different while still describing the same abstract group.