Subgroup

For other uses, see Subgroup (disambiguation).

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.

G is the group ${\displaystyle \mathbb {Z} /8\mathbb {Z} }$, the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$. 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

Main articles: Coset and Lagrange's theorem (group theory)

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,

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

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

+	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

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

The symmetric group S4 showing all permutations of 4 elements

All 30 subgroups Simplified Hasse diagrams of the lattice of subgroups of S4

12 elements

The alternating group A₄ showing only the even permutations

Subgroups:

8 elements

Dihedral group of order 8 Subgroups:

Dihedral group of order 8 Subgroups:

Dihedral group of order 8 Subgroups:

6 elements

Symmetric group S3 Subgroup:

Symmetric group S3 Subgroup:

Symmetric group S3 Subgroup:

Symmetric group S3 Subgroup:

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.

See also

• Cartan subgroup
• Fitting subgroup
• Fixed-point subgroup
• Fully normalized subgroup
• Stable subgroup

Notes

1. Gallian 2013, p. 61.
2. Hungerford 1974, p. 32.
3. Artin 2011, p. 43.
4. Kurzweil & Stellmacher 1998, p. 4.
5. Jacobson 2009, p. 41.
6. Ash 2002.
7. See a didactic proof in this video.
8. Dummit & Foote 2004, p. 90.
9. 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.