Groups, Basic Definitions and Cosets

Groups, Basic Definitions and Cosets

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 20 April 2010

Basic Definitions

We start with some basics, just a set and one operation. We can think of the operation as addition or multiplication, or something else, like a composition of functions.

A group is a set G and an operation × : G × G G (we write × ab as ab for abG ) such that

  1. g 1 g 2 g 3 = g 1 g 2 g 3 for all g 1 , g 2 , g 3 G.
  2. There exists an identity element, 1 G such that 1g=g1=g for all gG.
  3. For each gG there exists an inverse g -1 G of g such that g g -1 = g -1 =1.

A subgroup of a group G is a subset H G such that

  1. If h 1 , h 2 H   then   h 1 h 2 H
  2. 1H
  3. If h H   then   h -1 H.

The trivial group 1 is the set containing only 1 with the operation given by 1.1=1.

HW: Show that if G is a group, then the identity element of G is unique.

HW: Show that if gG then the inverse g -1 is unique.

HW: Why isn't 12345 a group?

Given such a definition the next step is to find out what kinds of structures fit the definition and explore them. Examples of groups are

  1. The integers with the operation of addition
  2. The integers mod n,   n , where n
  3. The symmetric group S n
  4. The general linear group of invertible matrices GL n .


Let G be a group and set H be a subgroup of G . We will use the subgroup H to divide up the group G .

  1. A left coset of H in G is a set gH= gh | hH where gG
  2. G/H (pronounced as G   mod H ) is the set of left cosets of H in G
  3. A right coset in G is a set Hg= hg | h H where gG
  4. H/G is the set of right cosets of H in G.

Unless we specify otherwise we shall always work with left cosets and just call them cosets.

HW: Let G be a group and let H be a subgroup of G . Let x and g be two elements of G. Show that x gH iff gH=xH.

g 1 H g 4 H g 5 H g 6 H g 7 H g 8 H H g 3 H g 2 H G

Let G be a group and let H be a subgroup of G. Then the cosets of H in G partition G.


To show:

  1. If g G then g g' H for some g' G.
  2. If g1 H g2 H then g1 H= g2 H.
  3. (Proof) Let g G .
  4. Then g=g· 1 gH since 1 H.
  5. So g gH.
  6. (Proof) Assume g1 H g2 H .
  7. To show:
    1. g1 H g2 H .
    2. g2 H g1 H .
  8. Let k g1H g2 H.
  9. Suppose k= g1 h1 = g2 h2 , where h1 , h2 H.
  10. Then g1 = g1 h1 h1 -1 =k h1 -1 = g2 h2 h1 -1 , and g2 = g2 h2 h2 -1 =k h2 -1 = g1 h1 h2 -1 .
    1. (Proof) Let g g1 H.
    2. Then g= g1 h for some h H.
    3. Then g= g1 h= g2 h2 h1 -1 h g2 H, since h2 h1 -1 h H.
    4. So g1 H g2 H.
    5. (Proof) Let g g2 H.
    6. Then g= g2 h for some h H.
    7. Then g= g2 h= g1 h1 h2 -1 h g1 H, since h1 h2 -1 h H.
    8. So g2 H g1 H.
  11. So g1 H= g2 H.
So the cosets of H in G partition G.

Let G be a group and let H be a subgroup of G. Then, for any g 1 , g 2 G, Card g 1 H =Card g 2 H .


To show: There is a bijection from g1 H to g2 H .
Define a map φ by φ : g1 H g2 H x g2 g1 -1 x . To show:

  1. φ is well defined.
  2. φ is a bijection.
  3. (Proof) To show:
    1. If x g1 H then φ x g2 H.
    2. If x=y then φ x =φ y .
    3. (Proof) Assume x g1 H.
    4. Then x= g1 h for some h H.
    5. So φ x = g2 g1 -1 g1 h = g2 h g2 H.
    6. This is clear from the definition of φ .
    So φ is well defined.
  4. By virtue of Theorem 2.2.3 Part I, ???????????????????? we want to construct an inverse map for φ . Define ψ : g2 H g1 H y g1 g2 -1 y. HW: Show (exactly as in a) above) that ψ is well defined. Then ψ φ x = g1 g2 -1 φ x = g1 g2 -1 g2 g1 -1 x=x, and φ ψ y = g2 g1 -1 φ y = g2 g1 -1 g1 g2 -1 y=y. So ψ is an inverse function to φ .
    So φ is a bijection.

Let H be a subgroup of a group G. Then Card G =Card G/H Card H .


By Proposition 2.2, all cosets in G/ H are the same size as H .
Since the cosets of H partition G , the cosets are disjoint subsets of G, and G is a union of those subsets.
So G is the union of Card G/ H disjoint subsets all of which have size Card H .

The above results show that the cosets of a subgroup H divide the group G into equal size pieces, one of these pieces being the subgroup H itself.

A set of coset representatives of H in G is a set of distinct elements g i of G such that

  1. each coset of H is of the form g i H for some g i and
  2. g i H g j H unless g i = g j .

The index of a subgroup H in a group G,   G:H is the number of cosets of H in G. G:H =Card G/H .

HW: Show that G: 1 =Card G .

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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