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 Template:Mvar under a binary operation ∗, a subset Template:Mvar of Template:Mvar is called a subgroup of Template:Mvar if Template:Mvar also forms a group under the operation ∗. More precisely, Template:Mvar is a subgroup of Template:Mvar if the restriction of ∗ to H × H is a group operation on Template:Mvar. This is often denoted H ≤ G, read as "Template:Mvar is a subgroup of Template:Mvar".

The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.Template:Sfn

A proper subgroup of a group Template:Mvar is a subgroup Template:Mvar which is a proper subset of Template:Mvar (that is, H ≠ G). This is often represented notationally by H < G, read as "Template:Mvar is a proper subgroup of Template:Mvar". Some authors also exclude the trivial group from being proper (that is, H ≠ {e}Template:0ws).Template:SfnTemplate:Sfn

If Template:Mvar is a subgroup of Template:Mvar, then Template:Mvar is sometimes called an overgroup of Template:Mvar.

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

Subgroup tests

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

• Then Template:Mvar is a subgroup of Template:Mvar if and only if Template:Mvar is nonempty and closed under products and inverses. Closed under products means that for every Template:Mvar and Template:Mvar in Template:Mvar, the product Template:Mvar is in Template:Mvar. Closed under inverses means that for every Template:Mvar in Template:Mvar, the inverse a⁻¹ is in Template:Mvar. These two conditions can be combined into one, that for every Template:Mvar and Template:Mvar in Template:Mvar, the element ab⁻¹ is in Template:Mvar, but it is more natural and usually just as easy to test the two closure conditions separately.Template:Sfn
• When Template:Mvar is finite, the test can be simplified: Template:Mvar is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element Template:Mvar of Template:Mvar generates a finite cyclic subgroup of Template:Mvar, say of order Template:Mvar, and then the inverse of Template:Mvar is aⁿ⁻¹.Template:Sfn

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

Basic properties of subgroups

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

File:Left cosets of Z 2 in Z 8.svg

Template:Mvar is the group $\mathbb{\mathbb{Z}}/8\mathbb{\mathbb{Z}},$ the integers mod 8 under addition. The subgroup Template:Mvar contains only 0 and 4, and is isomorphic to $\mathbb{\mathbb{Z}}/2\mathbb{\mathbb{Z}}.$ There are four left cosets of Template:Mvar: Template:Mvar itself, 1 + H, 2 + H, and 3 + H (written using additive notation since this is an additive group). Together they partition the entire group Template:Mvar into equal-size, non-overlapping sets. The index [G : H] is 4.

Cosets and Lagrange's theorem

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

Lagrange's theorem states that for a finite group Template:Mvar and a subgroup Template:Mvar,

$[G:H]=\frac{|G|}{|H|}$

where Template:Mvar and Template:Mvar denote the orders of Template:Mvar and Template:Mvar, respectively. In particular, the order of every subgroup of Template:Mvar (and the order of every element of Template:Mvar) must be a divisor of Template:Mvar.Template:Sfn

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 Template:Mvar in Template:Mvar, then Template:Mvar 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 Template:Mvar is the lowest prime dividing the order of a finite group Template:Mvar, then any subgroup of index Template:Mvar (if such exists) is normal.

Example: Subgroups of Z₈

Let Template:Mvar be the cyclic group Z₈ whose elements are

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

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: Template:Colorbull J = {0, 4} and Template:Colorbull H = {0, 4, 2, 6} , where Template:Mvar is also a subgroup of Template:Mvar. The Cayley table for Template:Mvar is the top-left quadrant of the Cayley table for Template:Mvar; The Cayley table for Template:Mvar is the top-left quadrant of the Cayley table for Template:Mvar. The group Template:Mvar is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.Template:Sfn

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.

File:Symmetric group 4; Cayley table; numbers.svg Symmetric group 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.)

File:Alternating group 4; Cayley table; numbers.svg

Alternating group A₄

Subgroups:
File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg
File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4).svgFile:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg

8 elements

File:Dihedral group of order 8; Cayley table (element orders 1,2,2,2,2,4,4,2); subgroup of S4.svg Dihedral group of order 8 Subgroups: File:Klein four-group; Cayley table; subgroup of S4 (elements 0,1,6,7).svgFile:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svgFile:Cyclic group 4; Cayley table (element orders 1,2,4,4); subgroup of S4.svg

File:Dihedral group of order 8; Cayley table (element orders 1,2,2,4,2,2,4,2); subgroup of S4.svg Dihedral group of order 8 Subgroups: File:Klein four-group; Cayley table; subgroup of S4 (elements 0,5,14,16).svgFile:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svgFile:Cyclic group 4; Cayley table (element orders 1,4,2,4); subgroup of S4.svg

File:Dihedral group of order 8; Cayley table (element orders 1,2,2,4,4,2,2,2); subgroup of S4.svg Dihedral group of order 8 Subgroups: File:Klein four-group; Cayley table; subgroup of S4 (elements 0,2,21,23).svgFile:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svgFile:Cyclic group 4; Cayley table (element orders 1,4,4,2); subgroup of S4.svg

6 elements

File:Symmetric group 3; Cayley table; subgroup of S4 (elements 0,1,2,3,4,5).svg Symmetric group S3 Subgroup:File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4).svg

File:Symmetric group 3; Cayley table; subgroup of S4 (elements 0,5,6,11,19,21).svg Symmetric group S3 Subgroup:File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg

File:Symmetric group 3; Cayley table; subgroup of S4 (elements 0,1,14,15,20,21).svg Symmetric group S3 Subgroup:File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg

File:Symmetric group 3; Cayley table; subgroup of S4 (elements 0,2,6,8,12,14).svg Symmetric group S3 Subgroup:File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg

4 elements

File:Klein four-group; Cayley table; subgroup of S4 (elements 0,1,6,7).svg Klein four-group

File:Klein four-group; Cayley table; subgroup of S4 (elements 0,5,14,16).svg Klein four-group

File:Klein four-group; Cayley table; subgroup of S4 (elements 0,2,21,23).svg Klein four-group

File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg Klein four-group (normal subgroup)

File:Cyclic group 4; Cayley table (element orders 1,2,4,4); subgroup of S4.svg Cyclic group Z4

File:Cyclic group 4; Cayley table (element orders 1,4,2,4); subgroup of S4.svg Cyclic group Z4

File:Cyclic group 4; Cayley table (element orders 1,4,4,2); subgroup of S4.svg Cyclic group Z4

3 elements

File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4).svg Cyclic group Z3

File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg Cyclic group Z3

File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg Cyclic group Z3

File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg Cyclic group Z3

2 elements

Each permutation Template:Mvar 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 Template:Tmath of the integer ring Template:Tmath the sum of two even integers is even, and the negative of an even integer is even.
• An ideal in a ring Template:Mvar is a subgroup of the additive group of Template:Mvar.
• 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

References

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