Two groups are essentially the same group if they have the same group structure. If two groups have the same group structure which can find a mapping between the groups, called an isomorphism, which preserver the structure of the groups, including the order of each element, the relationships between the elements, inverses and identities.
Classifying a group means classifying the group up to isomorphism. To classify groups of order 4, we can start by looking at the possible orders of the elements. The only numbers which divide 4 are 4, 2, 1. If the group has an element a of order 4, it is cyclic and is isomorphic to
If there is no element of order 4 there must be three elements of order 2. There can only be one element of order 1 – the identity. Any element of order 2 is its own inverse so the group is abelian ( if
then
so
but
and
so
) and
for every
If three elements of the group are
the fourth element must be
and this must be equal to
Neither of these can be equal to
since if
then
but since
is self inverse,
contradicting the fact that the inverse of an element is unique. The group is isomorphic to the group of symmetries of the rectangle,
also called the Klein group.
There are no other groups of order 4.