Subgroup

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 H ≤ 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 ≠ 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 ≠ {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 aⁿ⁻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 → 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 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 A ⊆ B or B ⊆ A. 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 : H → 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 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,

[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

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

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, S₄ is a subgroup of itself.

All 30 subgroups

Simplified

Hasse diagrams of the lattice of subgroups of S₄

12 elements

The alternating group contains only the even permutations.
It is one of the two nontrivial proper normal subgroups of S₄. (The other one is its Klein subgroup.)

8 elements

6 elements

4 elements

3 elements

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 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 $R$ 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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