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 $G$ 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 $H \times H$ is a group operation on $H$ . This is often denoted $H \leq G$ , read as " $H$ is a subgroup of $G$ ".

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, $H \neq G$ ). 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, $H \neq {e}$ ).^[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 $a -1$ is in $H$ . These two conditions can be combined into one, that for every $a$ and $b$ in $H$ , the element $ab -1$ 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 $a n -1$ .^[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 $e G$ , and $H$ is a subgroup of $G$ with identity $e H$ , then $e H = e G$ .
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 $ab = ba = e H$ , then $ab = ba = e G$ .
If $H$ is a subgroup of $G$ , then the inclusion map $H \to G$ 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 ⁠ $\mathbb {R} ^{2}$ ⁠ 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 $A \subseteq B$ or $B \subseteq A$ . A non-example: ⁠ $2\mathbb {Z} \cup 3\mathbb {Z}$ ⁠ is not a subgroup of ⁠ $\mathbb {Z} ,$ ⁠ 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 ⁠ $\mathbb {R} ^{2}$ ⁠ is not a subgroup of ⁠ $\mathbb {R} ^{2}.$ ⁠
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 ⁠ $\mathbb {Z} /n\mathbb {Z}$ ⁠ (the integers $mod n$ ) for some positive integer $n$ , then $n$ is the smallest positive integer for which $a n = e$ , and $n$ is called the order of $a$ . If $⟨ a ⟩$ is isomorphic to ⁠ $\mathbb {Z} ,$ ⁠ 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.

G

is the group

\mathbb {Z} /8\mathbb {Z} ,

the integers mod 8 under addition. The subgroup

H

contains only 0 and 4, and is isomorphic to

\mathbb {Z} /2\mathbb {Z} .

There are four left cosets of

H

:

H

itself,

1 + H

,

2 + H

, and

3 + H

(written using additive notation since this is an additive group). Together they partition the entire group

G

into equal-size, non-overlapping sets. The index

[G : H]

is 4.

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 $φ : H \to aH$ 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 $a 1 ~ a 2$ if and only if ⁠ $a_{1}^{-1}a_{2}$ ⁠ 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 $H$ ,

[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 $Z 8$ 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: $■ J = {0, 4}$ 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

$S 4$ 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, $S 4$ is a subgroup of itself.

Symmetric group

S 4

All 30 subgroups

Simplified

Hasse diagrams of the lattice of subgroups of

S 4

12 elements

The alternating group contains only the even permutations.
It is one of the two nontrivial proper normal subgroups of $S 4$ . (The other one is its Klein subgroup.)

Alternating group

A 4

Subgroups:

8 elements

Dihedral group of order 8

Subgroups:

6 elements

Symmetric group

S 3

Subgroup:

Symmetric group

S 3

Subgroup:

Symmetric group

S 3

Subgroup:

Symmetric group

S 3

Subgroup:

4 elements

Klein four-group

Klein four-group
(normal subgroup)

Cyclic group

Z 4

Cyclic group

Z 4

Cyclic group

Z 4

3 elements

Cyclic group

Z 3

Cyclic group

Z 3

Cyclic group

Z 3

Cyclic group

Z 3

2 elements

Each permutation $p$ of order 2 generates a subgroup ${1, p$ }. 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 ⁠ $2\mathbb {Z}$ ⁠ of the integer ring ⁠ $\mathbb {Z} :$ ⁠ the sum of two even integers is even, and the negative of an even integer is even.
An ideal in a ring $R$ is a subgroup of the additive group of $R$ .
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

^ Gallian 2013, p. 61.
^ Hungerford 1974, p. 32.
^ Artin 2011, p. 43.
^ ^a ^b Kurzweil & Stellmacher 1998, p. 4.
^ Jacobson 2009, p. 41.
^ Ash 2002.
^ See a didactic proof in this video.
^ Dummit & Foote 2004, p. 90.
^ Gallian 2013, p. 81.

References

Jacobson, Nathan (2009), Basic algebra, vol. 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1.
Hungerford, Thomas (1974), Algebra (1st ed.), Springer-Verlag, ISBN 9780387905181.
Artin, Michael (2011), Algebra (2nd ed.), Prentice Hall, ISBN 9780132413770.
Dummit, David S.; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 9780471452348. OCLC 248917264.
Gallian, Joseph A. (2013). Contemporary abstract algebra (8th ed.). Boston, MA: Brooks/Cole Cengage Learning. ISBN 978-1-133-59970-8. OCLC 807255720.
Kurzweil, Hans; Stellmacher, Bernd (1998). Theorie der endlichen Gruppen. Springer-Lehrbuch. doi:10.1007/978-3-642-58816-7.
Ash, Robert B. (2002). Abstract Algebra: The Basic Graduate Year. Department of Mathematics University of Illinois.

[FOOTNOTEGallian201361-1] Gallian 2013, p. 61.

[FOOTNOTEHungerford197432-2] Hungerford 1974, p. 32.

[FOOTNOTEArtin201143-3] Artin 2011, p. 43.

[FOOTNOTEKurzweilStellmacher19984-4] Kurzweil & Stellmacher 1998, p. 4.

[FOOTNOTEJacobson200941-5] Jacobson 2009, p. 41.

[FOOTNOTEAsh2002-6] Ash 2002.

[7] See a didactic proof in this video.

[FOOTNOTEDummitFoote200490-8] Dummit & Foote 2004, p. 90.

[FOOTNOTEGallian201381-9] Gallian 2013, p. 81.

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Subgroup

Contents

Subgroup tests

Basic properties of subgroups

Cosets and Lagrange's theorem

Example: Subgroups of Z₈