Group Actions

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: 9 February 2010

Group Actions

An actionof a group $G$ on a set $S$ is a mapping $α : G \times S \to S$ (the convention is to write $g s$ for α gs ) such that

$g (h s) = (g h) s$ for all $g, h \in G, s \in S .$
$1 s = s$ for all $s \in S .$

Examples of group actions are given below in this section and in the Exercises.

Suppose a group $G$ , a set $S$ and an action of $G$ on $S$ are given.

The stabiliser of an element $s \in S$ under the action of $G$ is the set $G_{s} = \{g \in G | g s = s\} .$

The orbit of an element $s \in S$ under the action of $G$ is the set $G s = \{s' \in S | g s = s' for some g \in G\} .$

Suppose $G$ is a group acting on a set $S$ and let $s \in S$ and $g \in G .$ Then

$G_{s}$ is a subgroup of $G .$
$G_{g s} = g G_{s} g^{- 1}$

The following is an analogue of Proposition 1.1.3

Let $G$ be a group which acts on a set $S .$ Then the orbits partition the set $S .$

If $G$ is a group acting on a set $S$ and $G_{s_{i}}$ denote the orbits of the action of $G$ on $S$ then $Card (S) = \sum_{distinct orbits} Card (G_{s_{i}}) .$

It is possible to view the stabiliser $G_{s}$ of an element $s \in S$ as an analogue of the kernel of a homomorphism and the orbit $G s$ of an element $s \in S$ as an analogue of the image of a homomorphism. One might say $\begin{array}{lcl} group actions α : G \times S \to S & are to & group homomorphisms f : G \to H & as \\ stabilisers G_{s} & are to & kernels \ker f & as \\ orbits G s & are to & images im f . \end{array}$

From this point of view the following corollary is an analogue of Corollary 1.1.5.

Let $G$ be a group acting on a set $S$ and let $s \in S .$ If $G s$ is the orbit containing $s$ and $G_{s}$ is the stabiliser of $s$ then $|G : G_{s}| = Card (G s)$ where $|G : G_{s}|$ is the index of $G_{s} \in G .$

Let $G$ be a group acting on a set $S .$ Let $G_{s}$ denote the stabiliser of $s$ and let $G s$ denote the orbit of $s .$ Then $Card (G) = Card (G s) Card (G_{s}) .$

Conjugation

Let $S$ be a subset of a group $G .$ The normaliser of $S$ in $G$ is the set $N_{S} = \{x \in G | x S x^{- 1} = S\},$ where $x S x^{- 1} = \{x s x^{- 1} | s \in S\} .$

Let $H$ be a subgroup of $G$ and let $N_{H}$ be the normaliser of $H$ in $G .$ Then

$H$ is a normal subgroup of $N_{H} .$
If $K$ is a subgroup of $G$ such that $H \subseteq K \subseteq G$ and $H$ is a normal subgroup of $K$ then $K \subseteq N_{H} .$

This proposition says that $N_{H}$ is the largest ubgroup of $G$ such that $H$ is normal in this subgroup.

Let $G$ be a group and let $𝒮$ be the set of subsets of $G .$ Then

$G$ acts on $𝒮$ by $\begin{array}{rcl} G \times 𝒮 & \to & 𝒮 \\ (g, S) & \to & g S g^{- 1} \end{array}$ where $g S g^{- 1} = \{g s g^{- 1} | s \in S\} .$ We say that $G$ acts on $𝒮$ by conjugation.
If $S$ is a subset of $G$ then $N_{S}$ is the stabiliser of $S$ under the ction of $G$ on $𝒮$ by conjugation.

Two elementd $g_{1}, g_{2} \in G$ are said to be conjugate if $g_{1} = h g_{2} h^{- 1}$ for some $h \in G .$

Let $G$ be a group and let $g \in G .$ The conjugacy class $𝒞_{g}$ of $g$ is the set of all conjugates of $g .$

Let $g$ be an element of a group $G .$ The centraliser or normaliser of $g$ is the set $Z_{g} = \{x \in G | x g x^{- 1} = g\} .$

Let $G$ be a group. Then

$G$ acts on $G$ by $\begin{array}{rcl} G \times G & \to & G \\ (g, s) & \mapsto & g s g^{- 1} . \end{array}$ We say that $G$ acts on itself by conjugation.
Two elements $g_{1}, g_{2} \in G$ are conjugate iff they are in the same orbit under the action of $G$ on itself via conjugation.
The conjugacy class $𝒞_{g}$ of $g \in G$ is the orbit of $g$ under the action of $G$ on itself via conjugation.
The centraliser $Z_{g}$ of $g \in G$ is the stabiliser of $g \in G$ under the action of $G$ on itself via conjugation.

Let $S$ be a subset of a group $G .$ The centraliser of $S$ in $G$ is the set $Z_{S} = \{x \in G | x s x^{- 1} for all s \in S\} .$

Let $G_{s}$ be the stabiliser of $s \in G$ under the action of $G$ on itself by conjugation. Then

For each subset $S \subseteq G,$ $Z_{S} = \underset{s \in S}{\cap} G_{s} .$
$Z (G) = Z_{G},$ where $Z (G)$ denotes the center of $G .$
$s \in Z (G)$ iff $Z_{s} = G .$
$s \in Z (G)$ iff $𝒞_{s} = \{s\} .$

(The Class Equation) Let $𝒞_{g_{i}}$ denote the conjugacy classes in a group $G$ and let $|𝒞_{g_{i}}|$ denote $Card (𝒞_{g_{i}}) .$ Then $|G| = |Z (G)| + \sum_{|Z (G)| > 1} Card (𝒞_{g_{i}}) .$

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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