Subgroup

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Subset of a group that forms a group itself

title: "Subgroup" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["group-theory", "subgroup-properties"] description: "Subset of a group that forms a group itself" topic_path: "general/group-theory" source: "https://en.wikipedia.org/wiki/Subgroup" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Subset of a group that forms a group itself ::

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 × 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.

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 {{math|H

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.
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. 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 eG, and H is a subgroup of G with identity eH, then .
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 , then .
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. For example, the intersection of the x-axis and y-axis in 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: is not a subgroup of 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 is not a subgroup of
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 and is called the subgroup generated by S. An element of G is in if and only if it is a finite product of elements of S and their inverses, possibly repeated.
Every element a of a group G generates a cyclic subgroup . If is isomorphic to (the integers mod n) for some positive integer n, then n is the smallest positive integer for which , and n is called the order of a. If is isomorphic to 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.

::figure[src="https://upload.wikimedia.org/wikipedia/commons/d/d4/Left_cosets_of_Z_2_in_Z_8.svg" caption="[''G'' : ''H'']}} is 4."] ::

Cosets and Lagrange's theorem

Main article: Coset, Lagrange's theorem (group theory)

Given a subgroup H and some a in G, we define the left coset Because a is invertible, the map φ : H → aH given by 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 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 and 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 .

Right cosets are defined analogously: They are also the equivalence classes for a suitable equivalence relation and their number is equal to [G : H].

If 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

::data[format=table]

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

This group has two nontrivial subgroups: and , 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.

Example: Subgroups of S₄{{anchor|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.

::data[format=table] | [[File:Symmetric group 4; Cayley table; numbers.svg|thumb|left|595px|Symmetric group S4]] | | align = right | image1 = Symmetric group S4; lattice of subgroups Hasse diagram; all 30 subgroups.svg | width1 = 250 | caption1 = All 30 subgroups | image2 = Symmetric group S4; lattice of subgroups Hasse diagram; 11 different cycle graphs.svg | width2 = 185 | caption2 = Simplified | 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.) ::figure[src="https://upload.wikimedia.org/wikipedia/commons/8/8b/Alternating_group_4;_Cayley_table;_numbers.svg" caption="60px]] [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg"] ::

8 elements

Subgroups: [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,1,6,7).svg|70px]][[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|70px]][[File:Cyclic group 4; Cayley table (element orders 1,2,4,4); subgroup of S4.svg|70px]]]] | | [[File:Dihedral group of order 8; Cayley table (element orders 1,2,2,4,2,2,4,2); subgroup of S4.svg|thumb|233px|Dihedral group of order 8

Subgroups: [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,5,14,16).svg|70px]][[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|70px]][[File:Cyclic group 4; Cayley table (element orders 1,4,2,4); subgroup of S4.svg|70px]]]] | | [[File:Dihedral group of order 8; Cayley table (element orders 1,2,2,4,4,2,2,2); subgroup of S4.svg|thumb|233px|Dihedral group of order 8

Subgroups: [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,2,21,23).svg|70px]][[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|70px]][[File:Cyclic group 4; Cayley table (element orders 1,4,4,2); subgroup of S4.svg|70px]]]] | |---|---|---|---|---| ::

6 elements

Subgroup:[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg|60px]]]] | |---|---|---|---| ::

4 elements

::data[format=table] | [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,1,6,7).svg|thumb|142px|[[w:Klein four-group|Klein four-group]]]] | [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,5,14,16).svg|thumb|142px|Klein four-group]] | [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,2,21,23).svg|thumb|142px|Klein four-group]] | [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|thumb|142px|Klein four-group ([[normal subgroup]])]] | |---|---|---|---| ::

::data[format=table] | [[File:Cyclic group 4; Cayley table (element orders 1,2,4,4); subgroup of S4.svg|thumb|142px|[[w:Cyclic group|Cyclic group]] Z4]] | [[File:Cyclic group 4; Cayley table (element orders 1,4,2,4); subgroup of S4.svg|thumb|142px|Cyclic group Z4]] | [[File:Cyclic group 4; Cayley table (element orders 1,4,4,2); subgroup of S4.svg|thumb|142px|Cyclic group Z4]] | |---|---|---| ::

3 elements

::data[format=table] | [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4).svg|thumb|120px|[[w:Cyclic group|Cyclic group]] Z3]] | [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg|thumb|120px|Cyclic group Z3]] | [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg|thumb|120px|Cyclic group Z3]] | | |---|---|---|---| ::

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 of the integer ring 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

References

See a [https://www.youtube.com/watch?v=TCcSZEL_3CQ didactic proof in this video].

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::

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

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

Subgroup

Subgroup tests

Basic properties of subgroups

Cosets and Lagrange's theorem

Example: Subgroups of Z8==

Example: Subgroups of S4{{anchor|Subgroups of S4}}

24 elements

12 elements

8 elements

6 elements

4 elements

3 elements

2 elements

1 element

Other examples

Notes

References

References

Example: Subgroups of Z₈==

Example: Subgroups of S₄{{anchor|Subgroups of S4}}

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