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 {e} 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.

thumb| is the group <math>\Z/8\Z,</math> the [[Integers modulo n|integers mod 8 under addition. The subgroup contains only 0 and 4, and is isomorphic to <math>\Z/2\Z.</math> There are four left cosets of : itself, , , and (written using additive notation since this is an additive group). Together they partition the entire group into equal-size, non-overlapping sets. The index is 4.]]

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 ,

: <math> [ G : H ] = { |G| \over |H| }</math>

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 .

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 Z<sub>8</sub>==<!-- This section is linked from List of small groups -->

Let be the finite cyclic group

:<math>\mathrm{Z}_8 = \{0,1,2,3,4,5,6,7\}</math>

under addition modulo 8.

The subset <math>\{0,2,4,6\}</math> consisting of multiples of 2 is a subgroup of <math>\mathrm{Z}_8</math>.

More generally, for each divisor of 8, the multiples of form a subgroup.

Explicitly, for <math>d=1,2,4,8</math>, these subgroups are <math>\{0,1,2,3,4,5,6,7\}, \{0,2,4,6\}, \{0,4\}, \{0\}</math>.

In general, for any positive integer , one can describe all subgroups of the finite cyclic group <math>\mathrm{Z}_n</math> similarly: for each divisor of , the multiples of in <math>\mathrm{Z}_n</math> form a subgroup of order <math>n/d</math>, and every subgroup arises in this way.

Subgroups of cyclic groups are cyclic.

Example: Subgroups of S<sub>4</sub>

The symmetric group is the group whose elements are the permutations of <math>\{1,2,3,4\}</math>.<br>

Below are all its subgroups, ordered by cardinality.<br>

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24 elements

Like each group, is a subgroup of itself.

12 elements

The alternating group consists of all the even permutations in .

Since it is of index 2, it is a normal subgroup.

8 elements

There are three subgroups of order 8, each isomorphic to the dihedral group , the group of symmetries of a square.

Labeling the vertices of a square <math>1,2,3,4</math> clockwise lets one view as a subgroup of .

This subgroup is generated by the 90-degree clockwise rotation and by the reflection in the diagonal axis joining vertices 1 and 3; these are the permutations <math>(1234)</math> and <math>(24)</math>.

Up to symmetries of the square, there are three different ways to label the vertices of a square, distinguished by which pairs of numbers appear on opposite corners.

In the labeling above, 1 and 3 were opposite, and 2 and 4 were opposite; another choice has 1 and 4 opposite, and 2 and 3 opposite; the third choice has 1 and 2 opposite, and 3 and 4 opposite.

The three labelings give rise to three different subgroups of order 8 in , conjugate to each other, each isomorphic to .

6 elements

There are four subgroups of order 6, each isomorphic to .

Each is the stabilizer of one of the elements of <math>\{1,2,3,4\}</math>.

For example, the stabilizer of 4 is the group of permutations in that map 4 to 4, while permuting <math>\{1,2,3\}</math> in an arbitrary way; it is generated by the permutations <math>(12)</math> and <math>(123)</math>, for instance.

The four subgroups of order 6 are conjugate to each other.

4 elements

There are seven subgroups of order 4, falling into three conjugacy classes of subgroups:

  • The subset <math>\{1,(12)(34),(13)(24),(14)(23)\}</math> is a normal subgroup isomorphic to the Klein four-group .
  • The group generated by <math>(12)</math> and <math>(34)</math> is another subgroup isomorphic to , but it is not normal. Instead it has conjugates, namely the group generated by <math>(13)</math> and <math>(24)</math> and the group generated by <math>(14)</math> and <math>(23)</math>.
  • Each of the six 4-cycles in generates a cyclic subgroup of order 4, but each 4-cycle generates the same subgroup as its inverse, so there are only three distinct subgroups of this type. These three subgroups are conjugate to each other because all 4-cycles in are conjugate to each other.

3 elements

There are four subgroups of order 3, each generated by a 3-cycle.

There are eight 3-cycles in , but each generates the same subgroup as its inverse.

The resulting four subgroups are conjugate to each other.

2 elements

There are nine subgroups of order 2, falling into two conjugacy classes of subgroups:

  • Each of the <math>\binom{4}{2} = 6</math> transpositions (2-cycles) generates a subgroup of order 2. These six subgroups are conjugate.
  • Each of the double-transpositions <math>(12)(34)</math>, <math>(13)(24)</math>, <math>(14)(23)</math> generates a subgroup of order 2. These three subgroups are conjugate.

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.
  • Every ideal in a ring is a subgroup of the additive group of .
  • Every 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.

Notes