Groups, Basic Definitions and Cosets

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

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 $\times : G \times G \to G$ (we write $\times (a, b)$ as $a b$ for $a b \in G$ ) such that

$(g_{1} g_{2}) g_{3} = g_{1} (g_{2} g_{3})$ for all $g_{1}, g_{2}, g_{3} \in G .$
There exists an identity element, $1 \in G$ such that $1 g = g 1 = g$ for all $g \in G .$
For each $g \in G$ there exists an inverse $g^{- 1} \in G$ of $g$ such that $g g^{- 1} = g^{- 1} = 1 .$

A subgroup of a group $G$ is a subset $H \subseteq G$ such that

If $h_{1}, h_{2} \in H then h_{1} h_{2} \in H$
$1 \in H$
If $h \in H then h^{- 1} \in H .$

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 $g \in G$ then the inverse $g^{- 1}$ is unique.

HW: Why isn't $\{1, 2, 3, 4, 5\}$ 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

The integers $ℤ$ with the operation of addition
The integers mod $n, ℤ_{n},$ where $n \in ℕ$
The symmetric group $S_{n}$
The general linear group of invertible matrices ${GL}_{n} (ℂ) .$

Cosets

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

A left coset of $H$ in $G$ is a set $g H = \{g h | h \in H\}$ where $g \in G$
$G / H$ (pronounced as $G$ mod $H$ ) is the set of left cosets of $H$ in $G$
A right coset in $G$ is a set $H g = \{h g | h \in H\}$ where $g \in G$
$H / G$ 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 $x \in g H$ iff $g H = x H .$

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

Proof.

To show:

If $g \in G$ then $g \in g' H$ for some $g' \in G .$
If $g_{1} H \cap g_{2} H \neq \emptyset$ then $g_{1} H = g_{2} H .$
(Proof) Let $g \in G .$
Then $g = g \cdot 1 \in g H$ since $1 \in H .$
So $g \in g H .$
(Proof) Assume $g_{1} H \cap g_{2} H \neq \emptyset .$
To show:
1. $g_{1} H \subseteq g_{2} H .$
2. $g_{2} H \subseteq g_{1} H .$
Let $k \in g_{1} H \cap g_{2} H .$
Suppose $k = g_{1} h_{1} = g_{2} h_{2},$ where $h_{1}, h_{2} \in H .$
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 $g \in g_{1} H .$
2. Then $g = g_{1} h$ for some $h \in H .$
3. Then $g = g_{1} h = g_{2} h_{2} {h_{1}}^{- 1} h \in g_{2} H,$ since $h_{2} {h_{1}}^{- 1} h \in H .$
4. So $g_{1} H \subseteq g_{2} H .$
5. (Proof) Let $g \in g_{2} H .$
6. Then $g = g_{2} h$ for some $h \in H .$
7. Then $g = g_{2} h = g_{1} h_{1} {h_{2}}^{- 1} h \in g_{1} H,$ since $h_{1} {h_{2}}^{- 1} h \in H .$
8. So $g_{2} H \subseteq g_{1} H .$
So $g_{1} H = g_{2} H .$

So the cosets of

H

G

partition

G .

$□$

Let $G$ be a group and let $H$ be a subgroup of $G .$ Then, for any $g_{1}, g_{2} \in G,$ $Card (g_{1} H) = Card (g_{2} H) .$

Proof.

To show: There is a bijection from $g_{1} H$ to $g_{2} H .$
Define a map $φ$ by $\begin{array}{rcl} φ : & g_{1} H & \to & g_{2} H \\ x & \mapsto & g_{2} {g_{1}}^{- 1} x . \end{array}$ To show:

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

$□$

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

Proof.

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

each coset of $H$ is of the form $g_{i} H$ for some $g_{i}$ and
$g_{i} H \neq g_{j} H$ 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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