Subgroup Explained
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗. More precisely, is a subgroup of if the restriction of ∗ to is a group operation on . This is often denoted, read as " is a subgroup of ".
The trivial subgroup of any group is the subgroup consisting of just the identity element.
A proper subgroup of a group is a subgroup which is a proper subset of (that is,). This is often represented notationally by, read as " is a proper subgroup of ". Some authors also exclude the trivial group from being proper (that is,).
If is a subgroup of, then is sometimes called an overgroup of .
The same definitions apply more generally when is an arbitrary semigroup, but this article will only deal with subgroups of groups.
Subgroup tests
Suppose that is a group, and is a subset of . For now, assume that the group operation of is written multiplicatively, denoted by juxtaposition.
- Then is a subgroup of if and only if is nonempty and closed under products and inverses. Closed under products means that for every and in, the product is in . Closed under inverses means that for every in, the inverse is in . These two conditions can be combined into one, that for every and in, the element is in, but it is more natural and usually just as easy to test the two closure conditions separately.
- When is finite, the test can be simplified: is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element of generates a finite cyclic subgroup of, say of order, and then the inverse of is .
If the group operation is instead denoted by addition, then closed under products should be replaced by closed under addition, which is the condition that for every and in, the sum is in, and closed under inverses should be edited to say that for every in, the inverse is in .
Basic properties of subgroups
- The identity of a subgroup is the identity of the group: if is a group with identity, and is a subgroup of with identity, then .
- The inverse of an element in a subgroup is the inverse of the element in the group: if is a subgroup of a group, and and are elements of such that, then .
- If is a subgroup of, then the inclusion map sending each element of to itself is a homomorphism.
- The intersection of subgroups and of is again a subgroup of . For example, the intersection of the -axis and -axis in under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of is a subgroup of .
- The union of subgroups and is a subgroup if and only if or . A non-example: is not a subgroup of because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the -axis and the -axis in is not a subgroup of
- If is a subset of, then there exists a smallest subgroup containing, namely the intersection of all of subgroups containing ; it is denoted by and is called the subgroup generated by . An element of is in if and only if it is a finite product of elements of and their inverses, possibly repeated.
- Every element of a group generates a cyclic subgroup . If is isomorphic to (the integers) for some positive integer, then is the smallest positive integer for which, and is called the order of . If is isomorphic to then is said to have infinite order.
- The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (While the infimum here is the usual set-theoretic intersection, the supremum of a set of subgroups is the subgroup generated by the set-theoretic union of the subgroups, not the set-theoretic union itself.) If is the identity of, then the trivial group is the minimum subgroup of, while the maximum subgroup is the group itself.
Cosets and Lagrange's theorem
See main article: Coset and Lagrange's theorem (group theory). Given a subgroup and some in, we define the left coset Because is invertible, the map given by is a bijection. Furthermore, every element of is contained in precisely one left coset of ; the left cosets are the equivalence classes corresponding to the equivalence relation if and only if is in . The number of left cosets of is called the index of in and is denoted by .
Lagrange's theorem states that for a finite group and a subgroup,
where and denote the
orders of and, respectively. In particular, the order of every subgroup of (and the order of every element of) must be a
divisor of .
[1] Right cosets are defined analogously: They are also the equivalence classes for a suitable equivalence relation and their number is equal to .
If for every in, then is said to be a normal subgroup. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if is the lowest prime dividing the order of a finite group, then any subgroup of index (if such exists) is normal.
Example: Subgroups of Z8
Let be the cyclic group whose elements are
G=\left\{0,4,2,6,1,5,3,7\right\}
and whose group operation is
addition modulo 8. Its
Cayley table is
| + | 0 | 4 | 2 | 6 | 1 | 5 | 3 | 7 |
|---|
| 0 | 0 | 4 | 2 | 6 | 1 | 5 | 3 | 7 |
|---|
| 4 | 4 | 0 | 6 | 2 | 5 | 1 | 7 | 3 |
|---|
| 2 | 2 | 6 | 4 | 0 | 3 | 7 | 5 | 1 |
|---|
| 6 | 6 | 2 | 0 | 4 | 7 | 3 | 1 | 5 |
|---|
| 1 | 1 | 5 | 3 | 7 | 2 | 6 | 4 | 0 |
|---|
| 5 | 5 | 1 | 7 | 3 | 6 | 2 | 0 | 4 |
|---|
| 3 | 3 | 7 | 5 | 1 | 4 | 0 | 6 | 2 |
|---|
| 7 | 7 | 3 | 1 | 5 | 0 | 4 | 2 | 6 | |
|---|
This group has two nontrivial subgroups: and, where is also a subgroup of . The Cayley table for is the top-left quadrant of the Cayley table for ; The Cayley table for is the top-left quadrant of the Cayley table for . The group is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.
Example: Subgroups of S4
is the symmetric group whose elements correspond to the permutations of 4 elements.
Below are all its subgroups, ordered by cardinality.
Each group (except those of cardinality 1 and 2) is represented by its Cayley table.
24 elements
Like each group, is a subgroup of itself.
12 elements
The alternating group contains only the even permutations.
It is one of the two nontrivial proper normal subgroups of . (The other one is its Klein subgroup.)
8 elements
6 elements
4 elements
3 elements
2 elements
Each permutation of order 2 generates a subgroup .These are the permutations that have only 2-cycles:
- There are the 6 transpositions with one 2-cycle. (green background)
- And 3 permutations with two 2-cycles. (white background, bold numbers)
1 element
The trivial subgroup is the unique subgroup of order 1.
Other examples
- The even integers form a subgroup of the integer ring the sum of two even integers is even, and the negative of an even integer is even.
- An ideal in a ring is a subgroup of the additive group of .
- A linear subspace of a vector space is a subgroup of the additive group of vectors.
- In an abelian group, the elements of finite order form a subgroup called the torsion subgroup.
See also
Notes
- See a didactic proof in this video.
References
- .
- .
- .
- Book: Abstract algebra. Dummit. David S.. Foote. Richard M.. 2004. Wiley. 9780471452348. 3rd. Hoboken, NJ. 248917264.
- Book: Gallian, Joseph A. . Joseph Gallian. Contemporary abstract algebra . 2013 . Brooks/Cole Cengage Learning . 978-1-133-59970-8 . 8th . Boston, MA . 807255720.
- Book: Kurzweil, Hans. Stellmacher. Bernd. 1998. Theorie der endlichen Gruppen. Springer-Lehrbuch. 10.1007/978-3-642-58816-7.
- Book: Ash, Robert B. . Abstract Algebra: The Basic Graduate Year . 2002 . Department of Mathematics University of Illinois . en.
