Subgroup

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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 HG, read as "H is a subgroup of G".

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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, HG). 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.

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

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

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 : HaH 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,

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

and whose group operation is addition modulo 8. Its Cayley table is

+04261537
004261537
440625173
226403751
662047315
115372640
551736204
337514062
773150426

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.

Symmetric group S4 Symmetric group 4; Cayley table; numbers.svg
Symmetric group S4
Symmetric group S4; lattice of subgroups Hasse diagram; all 30 subgroups.svg
All 30 subgroups
Symmetric group S4; lattice of subgroups Hasse diagram; 11 different cycle graphs.svg
Simplified

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.)

Alternating group A4

Subgroups: Alternating group 4; Cayley table; numbers.svg
Alternating group A4

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

8 elements

Dihedral group of order 8

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

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

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

6 elements

Symmetric group S3

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

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

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

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

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

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

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

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

4 elements

Klein four-group Klein four-group; Cayley table; subgroup of S4 (elements 0,1,6,7).svg
Klein four-group
Klein four-group Klein four-group; Cayley table; subgroup of S4 (elements 0,5,14,16).svg
Klein four-group
Klein four-group Klein four-group; Cayley table; subgroup of S4 (elements 0,2,21,23).svg
Klein four-group
Klein four-group
(normal subgroup) Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg
Klein four-group
(normal subgroup)
Cyclic group Z4 Cyclic group 4; Cayley table (element orders 1,2,4,4); subgroup of S4.svg
Cyclic group Z4
Cyclic group Z4 Cyclic group 4; Cayley table (element orders 1,4,2,4); subgroup of S4.svg
Cyclic group Z4
Cyclic group Z4 Cyclic group 4; Cayley table (element orders 1,4,4,2); subgroup of S4.svg
Cyclic group Z4

3 elements

Cyclic group Z3 Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4).svg
Cyclic group Z3
Cyclic group Z3 Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg
Cyclic group Z3
Cyclic group Z3 Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg
Cyclic group Z3
Cyclic group Z3 Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg
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:

1 element

The trivial subgroup is the unique subgroup of order 1.

Other examples

See also

Notes

  1. Gallian 2013, p. 61.
  2. Hungerford 1974, p. 32.
  3. Artin 2011, p. 43.
  4. 1 2 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.

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