Subgroups

A subgroup is a subset of a group that is itself a group.
Not every subset of a group is a subgroup. To be a subgroup, the subset must satisfy all properties of a group: closure, associativity, identity and inverse.

Subgroup Test

A simple way to check whether a subset H of a group G is a subgroup is to use the Subgroup Test.
The Subgroup Test states that a non-empty subset H of a group G is a subgroup if and only if for any two elements a and b in H, the product a(b^(-1)) is in H.

The set of even integers, under the operation of addition, is a subgroup of the group of integers, as it follows all group rules.
The group of non-zero rational numbers under multiplication operation has a subgroup of all rational numbers whose absolute value is 1.

Every group is a subgroup of itself.
The criteria for a subset to be a subgroup are rigorous to ensure that subgroup structure mirrors that of the group.
Subset containing the identity element of the group along with the inverses of all its elements is a subgroup. This is an immediate consequence of the Subgroup Test.

If H is a subgroup of G, then a left coset is a set formed by multiplying each element of H by a fixed element g of G.
Similarly, a right coset is a set formed by multiplying each element of H by g on the right side, i.e., Hg.
Lagrange’s theorem is a major result regarding subgroups, which states: in a finite group G, the order (number of elements) of every subgroup H of G divides the order of G. It implies that the number of left cosets (or right sets) equals the order of G divided by the order of H.
Important consequence of Lagrange’s theorem is that the order of a group’s element always divides the order of the group.

Remember, understanding this concept fully is important as it’s a key part of understanding groups, a core aspect of further mathematics.