Groups: Subgroups

Understanding Groups and Subgroups

A group is a mathematical structure consisting of a set of elements equipped with an operation that combines two elements to form a third.
In a group, the operation must be associative, there must be an identity element, and each element must have an inverse.
A subgroup is a subset of a group that is itself a group, with the operation inherited from the larger group.
To check if a subset H of a group G is a subgroup, we verify three properties: the identity of G is in H, if a and b are in H then so is the operation a*b, and if a is in H then so is its inverse.
If H is a subgroup of G and a is in G, then the left coset aH is the set of all ah where h is in H. Right cosets are defined similarly.

Considering the group of integers under addition, the even integers form a subgroup.
In the group of non-zero real numbers under multiplication, the positive real numbers form a subgroup.
Matrix groups offer many opportunities to explore subgroups. For instance, the invertible upper triangular matrices form a subgroup of the group of all invertible matrices..

Lagrange’s theorem, crucial in group theory, states that the order of any finite group G (the number of elements in G) is a multiple of the order of any of its subgroups H (the number of elements in H).
The study of subgroups allows us to break up problems involving a large group into manageable problems involving its subgroups.
Normal subgroups and the related concept of quotient groups are fundamental when studying the structure of a group.

The concept of subgroups is extended in ring theory and field theory, where we study subrings and subfields.
In topology, every topological group (a group with a topology such that the group’s operations are continuous) has a special type of subgroups called closed subgroups.
Subgroup analysis is crucial in studying the symmetry of mathematical and physical systems.