proper subgroup

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

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 ,
: G : H =
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 .See
didactic proof in this video

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\$
and whose group operation is addition modulo 8. Its Cayley table is

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

* Cartan subgroup
* Fitting subgroup
* Fixed-point subgroup
* Fully normalized subgroup
* Stable subgroup

Notes

References

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