World Library  
Flag as Inappropriate
Email this Article

Subgroup

Article Id: WHEBN0000028011
Reproduction Date:

Title: Subgroup  
Author: World Heritage Encyclopedia
Language: English
Subject: Symmetric group, Cyclic group, Ping-pong lemma, Group (mathematics), Lie group
Collection: Group Theory, Subgroup Properties
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Subgroup

In 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 represented notationally by HG, read as "H is a subgroup of G".

A proper subgroup of a group G is a subgroup H which is a proper subset of G (i.e. HG). The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. 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.

This article will write ab for ab, as is usual.

Contents

  • Basic properties of subgroups 1
  • Cosets and Lagrange's theorem 2
  • Example: Subgroups of Z8 3
  • Example: Subgroups of S4(the symmetric group on 4 elements) 4
    • 12 elements 4.1
    • 8 elements 4.2
    • 6 elements 4.3
    • 4 elements 4.4
    • 3 elements 4.5
  • See also 5
  • Notes 6
  • References 7

Basic properties of subgroups

  • A subset H of the group G is a subgroup of G if and only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever a and b are in H, then ab and a−1 are also in H. These two conditions can be combined into one equivalent condition: whenever a and b are in H, then ab−1 is also in H.) In the case that H is finite, then H is a subgroup if and only if H is closed under products. (In this case, every element a of H generates a finite cyclic subgroup of H, and the inverse of a is then a−1 = an − 1, where n is the order of a.)
  • The above condition can be stated in terms of a homomorphism; that is, H is a subgroup of a group G if and only if H is a subset of G and there is an inclusion homomorphism (i.e., i(a) = a for every a) from H to G.
  • 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 eH = eG.
  • 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 = eH, then ab = ba = eG.
  • The intersection of subgroups A and B is again a subgroup.[1] The union of subgroups A and B is a subgroup if and only if either A or B contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not. Another example is the union of the x-axis and the y-axis in the plane (with the addition operation); each of these objects is a subgroup but their union is not. This also serves as an example of two subgroups, whose intersection is precisely the identity.
  • If S is a subset of G, then there exists a minimum subgroup containing S, which can be found by taking the intersection of all of subgroups containing S; it is denoted by <S> and is said to be 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.
  • Every element a of a group G generates the cyclic subgroup <a>. If <a> is isomorphic to Z/nZ 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 \mathbb{Z}/8\mathbb{Z}, the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to \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

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,

[ 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|.

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

G=\left\{0,2,4,6,1,3,5,7\right\}
and whose group operation is addition modulo eight. Its Cayley table is
+ 0 2 4 6 1 3 5 7
0 0 2 4 6 1 3 5 7
2 2 4 6 0 3 5 7 1
4 4 6 0 2 5 7 1 3
6 6 0 2 4 7 1 3 5
1 1 3 5 7 2 4 6 0
3 3 5 7 1 4 6 0 2
5 5 7 1 3 6 0 2 4
7 7 1 3 5 0 2 4 6

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

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:


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

Cyclic group Z4
Cyclic group Z4


3 elements

Cyclic group Z3
Cyclic group Z3
Cyclic group Z3


See also

Notes

  1. ^ Jacobson (2009), p. 41

References

  •  .
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.