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 usually 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.

A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, HG). This is usually 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}). [1] [2]

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. The group G is sometimes denoted by the ordered pair (G, ∗), usually to emphasize the operation ∗ when G carries multiple algebraic or other structures.

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|. [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

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

+02461357
002461357
224603571
446025713
660247135
113572460
335714602
557136024
771350246

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.

The symmetric group S4 showing all permutations of 4 elements Symmetric group 4; Cayley table; numbers.svg
The symmetric group S4 showing all permutations of 4 elements
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 A4 showing only the even permutations

Subgroups: Alternating group 4; Cayley table; numbers.svg
The alternating group A4 showing only the even permutations

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

Other examples

See also

Notes

  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.

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