Group Actions Exercises

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: 11 February 2010

Group Actions

1. Let $G$ be a group. $G$ acts on itself by left multiplication. $$\begin{array}{rcl}G\times G& \to & G\\ \left(g,h\right)& \mapsto & gh.\end{array}$$

There is a single orbit, $G.$ If $h\in G$ is the stabiliser of $h$ is $\left(1\right).$

2. Prove the following important theorem by completing the steps below:

(Cayley's Theorem) Let $G$ be a finite group and let $n=\left|G\right|.$ Then $G$ is isomorphic to a subgroup of the symmetric group $S_n$ on $n.$

Let $G$ be a finite group and let $n=\left|G\right|.$ Let $S_n$ denote the symmetric group on $n.$ For each $g\in G,$ define a map $$\begin{array}{rcll}{m}_g:& G& \to & G\\ & h& \mapsto & gh.\end{array}$$

1. Show that we can view $m_g\in S_n$ as a permutation of the elements of $G.$
2. Show that if $g_1,g_2\in G$ then $m_{g_1}\circ m_{g_2}=m_{g_1{g}_2},$ since $m_{g_1}\left(m_{g_2}\left(h\right)\right)=m_{g_1}\left(g_{2}h\right)=g_1{g}_{2}h=m_{g_1{g}_2}\left(h\right).$
3. Show that if $g\in G$ denotes the identity in $G$ then $m_1$ is the identity map on $G.$
4. Show that if $g\in G$ then $m_{g^{-1}}$ is the inverse of $m_g.$
5. Show that if $g,h\in G$ and $m_g=m_h$ then $g=m_g\left(1\right)=m_h\left(1\right)=h.$ Define a map $$\begin{array}{rcll}\phi :& G& \to & S_g\\ & g& \mapsto & m_g.\end{array}$$
6. Show that by b) above, $\phi$ is a homomorphism.
7. Show that by e) $\phi is\; injective.$
8. Using Theorem 1.1.15 c), conclude that $$G\simeq \mathrm{im}\phi \subseteq S_n.$$

3. Let $H$ be a subgroup of a group $G$. $H$ acts on $G$ by right multiplication $$\begin{array}{rcl}H\times G& \to & G\\ \left(h,g\right)& \mapsto & g{h}^{-1}.\end{array}$$

1. Show that if $g\in G$, then the orbit of $g$ under this action is the coset $gH.$ Thus, the orbits are the cosets $G/H.$
2. Show that the stabiliser of an element $g\in G$ is the group $\left(1\right).$
3. Using Proposition 1.2.4, give another proof of Proposition 1.1.3.
4. Using Corollary 1.2.7, show that $$\mathrm{Card}\left(H\right)=\mathrm{Card}\left(gH\right)\mathrm{Card}\left(\left(1\right)\right)=\mathrm{Card}\left(gH\right),$$ and give another proof of Proposition 1.1.4.

4. Let $H$ be a subgroup of a group $G.$ $G$ acts on $G/H$ by left multiplication. $$\begin{array}{rcl}G\times G/H& \to & G/H\\ \left(g,kH\right)& \mapsto & gkH.\end{array}$$

1. Show that there is one orbit under this action, $G/H.$
2. Show that the stabiliser of the identity coset $H$ is $H$ and the stabiliser of a coset $kH,k\in G$ is the group $kH{k}^{-1}.$
3. Use Corollary 1.2.7 to show that $$\mathrm{Card}\left(G\right)=\mathrm{Card}\left(G/H\right)\mathrm{Card}\left(H\right),$$ and thus give another proof of Corollary 1.1.5.

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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