Subgroup

Last updated May 14, 2020

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 usually denoted H ≤ G, read as "H is a subgroup of G".

Basic properties of subgroups

A subset H of the group G is a subgroup of G if and only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever a and b are in H, then ab and a⁻¹ are also in H. These two conditions can be combined into one equivalent condition: whenever a and b are in H, then ab⁻¹ is also in H.) In the case that H is finite, then H is a subgroup if and only if H is closed under products. (In this case, every element a of H generates a finite cyclic subgroup of H, and the inverse of a is then a⁻¹ = aⁿ⁻¹, where n is the order of a.)
The above condition can be stated in terms of a homomorphism; that is, H is a subgroup of a group G if and only if H is a subset of G and there is an inclusion homomorphism (that is, i(a) = a for every a) from H to G.
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.
The intersection of subgroups A and B is again a subgroup.^[3] The union of subgroups A and B is a subgroup if and only if either A or B contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not. Another example is the union of the x-axis and the y-axis in the plane (with the addition operation); each of these objects is a subgroup but their union is not. This also serves as an example of two subgroups, whose intersection is precisely the identity.
If S is a subset of G, then there exists a minimum subgroup containing S, which can be found by taking the intersection of all of subgroups containing S; it is denoted by ⟨S⟩ and is said to be 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.
Every element a of a group G generates the cyclic subgroup ⟨a⟩. If ⟨a⟩ is isomorphic to Z/nZ 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|.^[4]^[5]

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\{0,2,4,6,1,3,5,7\right\}

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

+	0	2	4	6	1	3	5	7
0	0	2	4	6	1	3	5	7
2	2	4	6	0	3	5	7	1
4	4	6	0	2	5	7	1	3
6	6	0	2	4	7	1	3	5
1	1	3	5	7	2	4	6	0
3	3	5	7	1	4	6	0	2
5	5	7	1	3	6	0	2	4
7	7	1	3	5	0	2	4	6

This group has two nontrivial subgroups: J={0,4} and H={0,2,4,6}, 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 group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.

Example: Subgroups of S₄(the symmetric group on 4 elements)

Every group has as many small subgroups as neutral elements on the main diagonal:

The trivial group and two-element groups Z₂. These small subgroups are not counted in the following list.

All 30 subgroups

Simplified

Hasse diagrams of the lattice of subgroups of S₄

12 elements

8 elements

6 elements

4 elements

3 elements

Other examples

The even integers are a subgroup of the additive group of integers: when you add two even numbers, you get an even number.
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.
Let $A$ be an abelian group; the elements of $A$ that have finite period form a subgroup of $A$ called the torsion subgroup of $A$ .

Notes

↑ Hungerford (1974), p. 32
↑ Artin (2011), p. 43
↑ Jacobson (2009), p. 41
↑ See a didactic proof in this video.
↑ S., Dummit, David (2004). Abstract algebra. Foote, Richard M., 1950- (3. ed.). Hoboken, NJ: Wiley. p. 90. ISBN 9780471452348. OCLC 248917264.

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References

Jacobson, Nathan (2009), Basic algebra, 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 .
S., Dummit, David (2004). Abstract algebra. Foote, Richard M., 1950- (3. ed.). Hoboken, NJ: Wiley. ISBN 9780471452348. OCLC 248917264.

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[1] Hungerford (1974), p. 32

[2] Artin (2011), p. 43

[3] Jacobson (2009), p. 41

[4] See a didactic proof in this video.

[5] S., Dummit, David (2004). Abstract algebra. Foote, Richard M., 1950- (3. ed.). Hoboken, NJ: Wiley. p. 90. ISBN 9780471452348. OCLC 248917264.

Subgroup

Contents

Basic properties of subgroups

Cosets and Lagrange's theorem

Example: Subgroups of Z₈