The set {1,-1,i,-i} is closed under multipilication.It contains the identity 1 and every element has an inverse.1 and -1 are inverses of themselves while i and -i are inverses of each other.The given set is a subset of the group
of symmetries of the square. Hence,this set is a subgroup of
.In fact,it is a cyclic group with i and -i as generators.
On the other hand the subset {1,-1,C,-C} is not a cyclic group.Closure can be easily verified.The identity 1 is in the set.Moreover,each element is its own inverse.Hence,the set is a subgroup of
.
These two subgroups are the only two group structures of order 4.The former is isomorphic to
while the latter is isomorphic to V,that is Klein-4 group.(In fact,it follows from the fundamental theorem of finitely generated abelian groups that V is isomorphic to
x
)
The other subset {1,-1} is a subgroup as it is isomorphic to
.
