# Subgroup

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In group theory, a branch of mathematics, given a group G under a binary operation  ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is often denoted HG, read as "H is a subgroup of G".

## Contents

The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. [1]

A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, HG). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e}). [2] [3]

If H is a subgroup of G, then G is sometimes called an overgroup of H.

The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups.

## Subgroup tests

Suppose that G is a group, and H is a subset of G. For now, assume that the group operation of G is written multiplicatively, denoted by juxtaposition.

• Then H is a subgroup of G if and only if H is nonempty and closed under products and inverses. Closed under products means that for every a and b in H, the product ab is in H. Closed under inverses means that for every a in H, the inverse a1 is in H. These two conditions can be combined into one, that for every a and b in H, the element ab1 is in H, but it is more natural and usually just as easy to test the two closure conditions separately. [4]
• When H is finite, the test can be simplified: H is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element a of H generates a finite cyclic subgroup of H, say of order n, and then the inverse of a is an1. [4]

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 a and b in H, the sum a+b is in H, and closed under inverses should be edited to say that for every a in H, the inverse −a is in H.

## Basic properties of subgroups

• The identity of a subgroup is the identity of the group: if G is a group with identity eG, and H is a subgroup of G with identity eH, then eH = eG.
• The inverse of an element in a subgroup is the inverse of the element in the group: if H is a subgroup of a group G, and a and b are elements of H such that ab = ba = eH, then ab = ba = eG.
• If H is a subgroup of G, then the inclusion map HG sending each element a of H to itself is a homomorphism.
• The intersection of subgroups A and B of G is again a subgroup of G. [5] For example, the intersection of the x-axis and y-axis in R2 under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of G is a subgroup of G.
• The union of subgroups A and B is a subgroup if and only if AB or BA. A non-example: 2Z ∪ 3Z is not a subgroup of Z, because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the x-axis and the y-axis in R2 is not a subgroup of R2.
• If S is a subset of G, then there exists a smallest subgroup containing S, namely the intersection of all of subgroups containing S; it is denoted by S and is called the subgroup generated by S. An element of G is in S if and only if it is a finite product of elements of S and their inverses, possibly repeated. [6]
• Every element a of a group G generates a cyclic subgroup a. If a is isomorphic to Z/nZ (the integers mod n) for some positive integer n, then n is the smallest positive integer for which an = e, and n is called the order of a. If a is isomorphic to Z, then a 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 e is the identity of G, then the trivial group {e} is the minimum subgroup of G, while the maximum subgroup is the group G itself.

## Cosets and Lagrange's theorem

Given a subgroup H and some a in G, we define the left coset aH = {ah : h in H}. Because a is invertible, the map φ : HaH given by φ(h) = ah is a bijection. Furthermore, every element of G is contained in precisely one left coset of H; the left cosets are the equivalence classes corresponding to the equivalence relation a1 ~ a2 if and only if a1−1a2 is in H. The number of left cosets of H is called the index of H in G and is denoted by [G : H].

Lagrange's theorem states that for a finite group G and a subgroup H,

${\displaystyle [G:H]={|G| \over |H|}}$

where |G| and |H| denote the orders of G and H, respectively. In particular, the order of every subgroup of G (and the order of every element of G) must be a divisor of |G|. [7] [8]

Right cosets are defined analogously: Ha = {ha : h in H}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [G : H].

If aH = Ha for every a in G, then H 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 p is the lowest prime dividing the order of a finite group G, then any subgroup of index p (if such exists) is normal.

## Example: Subgroups of Z8

Let G be the cyclic group Z8 whose elements are

${\displaystyle G=\left\{0,4,2,6,1,5,3,7\right\}}$

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

+04261537
004261537
440625173
226403751
662047315
115372640
551736204
337514062
773150426

This group has two nontrivial subgroups: J = {0, 4} and H = {0, 4, 2, 6} , where J is also a subgroup of H. The Cayley table for H is the top-left quadrant of the Cayley table for G; The Cayley table for J is the top-left quadrant of the Cayley table for H. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic. [9]

## Example: Subgroups of S4

Let S4 be the symmetric group on 4 elements. Below are all the subgroups of S4, listed according to the number of elements, in decreasing order.

### 24 elements

The whole group S4 is a subgroup of S4, of order 24. Its Cayley table is

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### 8 elements

 Dihedral group of order 8 Subgroups: Dihedral group of order 8Subgroups: Dihedral group of order 8Subgroups:

### 6 elements

 Symmetric group S3 Subgroup: Symmetric group S3Subgroup: Symmetric group S3Subgroup: Symmetric group S3Subgroup:

### 4 elements

 Klein four-group Klein four-group Klein four-group Klein four-group
 Cyclic group Z4 Cyclic group Z4 Cyclic group Z4

### 3 elements

 Cyclic group Z3 Cyclic group Z3 Cyclic group Z3 Cyclic group Z3

### 2 elements

Each element s of order 2 in S4 generates a subgroup {1,s} of order 2. There are 9 such elements: the ${\displaystyle {\binom {4}{2}}=6}$ transpositions (2-cycles) and the three elements (12)(34), (13)(24), (14)(23).

### 1 element

The trivial subgroup is the unique subgroup of order 1 in S4.

## Other examples

• The even integers form a subgroup 2Z of the integer ring Z: the sum of two even integers is even, and the negative of an even integer is even.
• An ideal in a ring ${\displaystyle R}$ is a subgroup of the additive group of ${\displaystyle R}$.
• 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.

## Notes

1. Gallian 2013, p. 61.
2. Hungerford 1974, p. 32.
3. Artin 2011, p. 43.
4. Jacobson 2009, p. 41.
5. Dummit & Foote 2004, p. 90.
6. Gallian 2013, p. 81.

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