Any group has at least two subgroups: the trivial subgroup and itself. It need not necessarily have any other subgroups however; for example, has no nontrivial proper subgroup.
- The group of integers equipped with addition is a subgroup of the real numbers equipped with addition; i.e. .
- The group of real matrices with determinant 1 is a subgroup of the group of invertible real matrices, both equipped with matrix multiplication. To prove this, it is necessary to prove closure, meaning that it must be shown that the product of two matrices with determinant 1 is another matrix with determinant 1, but this is immediate from the multiplicative property of the determinant. This group is denoted , and is often seen.
- The set of complex numbers with magnitude 1 is a subgroup of the nonzero complex numbers equipped with multiplication. It is known as the circle group as its elements form the unit circle.
Another useful example is the subgroup , the set of multiples of equipped with addition:
where is any positive integer. These are the only nontrivial subgroups of the set of integers, and they help establish some classical number theory including the concept of greatest common divisor and least common multiple.
The subgroups of the additive group of integers denoted are of the form for some positive integer .
Consider a subgroup of . Then 0 must be in , as 0 is the identity of . If does not contain any element other than 0, then is the trivial group. Otherwise, suppose contains another element . Either or will be positive, so contains a positive element. Let be the smallest positive integer in .
It is necessary to show that
- every integer multiple of is in ;
- no other numbers are in .
The first is easy to show: since is a group, the axiom of closure applies, so is in for any positive integer . The inverse of is , so as well. Hence any multiple of is in .
Now suppose some other number were in . Then can be written in the form (for example, by the division algorithm), and since is not a multiple of , . But , so , which violates the assumption that is the smallest positive integer in .
So consists precisely of the multiples of .
The above proof shows that any subgroup is equal to , where is the smallest integer in the subgroup. This gives the following corollary:
Let be defined as
i.e. the elements are the sums of an element in and an element in . This is a subgroup of . Thus
for some integer , where is the greatest common divisor of and .
This also shows Bezout's identity:
for any , there exist integers such that
since the elements of are for some integer , and the elements of are for some integer .
is named the greatest common divisor because
- divides and as are elements of , since , and the elements of are multiples of , i.e. is a common divisor of ;
- If divides and , it also divides since if then , i.e. there is no greater common divisor than .
Similarly, for some integer . It is known as the least common multiple of and . The reasons are analogous to the above analysis.
First, it is important to have a criterion for checking whether some subset is in fact a subgroup. In particular, while the axiom of associativity is inherited from , both closure and the existence of identity/inverses need to be manually checked; since a given subset might not be closed, it might not contain the identity, and the inverse of an element may not be in the subset. Hence, a subgroup must satisfy the following:
- For all , as well.
- For all , as well.
These can be combined into a single form:
- For all , as well.
For this reason, the identity of is necessarily the identity of as well.
Given a way to recognize subgroups, the next natural question is how to go about finding them. Fortunately, this is easy to accomplish:
Any subset of defines a subgroup formed by taking the intersection of all subgroups containing . It is called the subgroup generated by .
The case where is a single element is of particular interest. In this case, the subgroup generated by is a cyclic group, and its size is called the order of the element (which may be familiar from modular arithmetic).
Given a subgroup, the next natural question is to examine precisely how the subgroup "fits" into the original. Cosets provide a framework for analyzing this.
A left coset of a subgroup is the set , where is an element of . The set of all left cosets of w.r.t. is denoted by .
A right coset of a subgroup is the set , where is an element of . The set of all right cosets of w.r.t. is denoted by . In general, right cosets are not independently very interesting.
For example, consider the subgroup of (which is an additive group). The left cosets are
so . The right cosets are
This example provides motivation for some properties of cosets:
- Every left coset has the same number of elements.
- The left cosets partition ; i.e. each element of appears in exactly one left coset.
Taken together, these properties combine to form Lagrange's theorem:
where is the order of
Note that this also demonstrates that the order of a subgroup necessarily divides the order of the original group.
In the above example, it was also true that the left cosets and right cosets were identical; i.e. for any , . This is not generally the case, but when it occurs, the subgroup is called normal. Normal subgroups are used to generate quotient groups, as was in the above example. The subgroups of abelian groups are always normal.