Let be a group. is a subgroup of if it is a group that is closed and contained within . This is denoted as .
A proper subgroup is defined as a subgroup of that is not in itself . This is denoted as . Otherwise, we say is improper.

General Properties

(Fraleigh 5.14) A subset is a subgroup of if and only if the following hold:
- is closed under the binary operation of .
- If is the identity element of , then .
- , .
(Fraleigh e5.54, Fraleigh 7.4) The intersection of subgroups is also a subgroup. That is if
(Fraleigh 10.10) Lagrange’s Theorem If is a finite group and , we have that divides .
- Intuition: Every coset contains the same number of elements as , and the cosets partition . Therefore, divides

Cyclic Subgroups

A special subgroup is called the cyclic subgroup. Let , then is the cyclic subgroup containing . It is defined as.

We refer to as the generator.
- (Fraleigh 5.17) It is the smallest subgroup containing .
In general, the smallest subgroup of containing all is the subgroup generated by . If this subgroup is all of , then generates , and each element is called a generator. If the set of is finite, then is finitely generated.
(Fraleigh 7.6) The subgroup generated by has elements which are precisely those elements of that are finite products of integral powers of the , where powers of a fixed may occur several times in the product.
A Cayley Digraph is a Directed Graph where each vertex corresponds to an element of and each edge , corresponds to where is a generator of . We overlay multiple such edge types for each element in some generating set .
A Cayley Digraph necessarily and sufficiently satisfies the following properties.
- The digraph is necessarily connected because every linear equation in the group has a solution.
- The digraph has at most one arc from to because the solution is unique.
- Each vertex has exactly one arc of each type starting at and one arc of each type ending at because the products are unique.
- If two different sequences of arc types starting from lead to the same vertex , then those same arc types starting from any vertex will lead to .
  
  If , then

If then we define
The join of subgroups is denoted and is defined as the intersection of all subgroups of that contain .
- is the smallest subgroup of containing as well as both and .
(Fraleigh 37.8) If and are finite subgroups of , then
- Idea: Each element can be represented in the form for and as many times as there are elements in .
Subgroup Series