8 Actions Group 8 L Definitions 8 Examples how our has Up to study of largely groups been about the groups themselves understanding e How elements G have order K g many of What are the subgroups of G What are the quotients of G Is G cyclic Abelian etc Now we will turn our attention to how studying a can describe the group G symmetries of other X we see how sets Essentially will a elements can permute the of X group course we don't want G to Of permute the elements of X arbitrarily we somehow want the structure G of in this reflected permutation For instance how would like the you ee to elements identity G permute the of X If our definition is at all reasonable it should leave everything unchanged be It would also reasonable to expect that for EG X the g gz permuting by should be the same as product g.ge and then here permuting by gz by g we read left like functions right to Definition Let G be a and X be group a set An action of G on X is a map s h the Gx X X wi following properties L x u e for all XEX 2 x gigs X for all g g c G g gz and all see X to We write GTX indicate that G is on X acting 1 on Ex The group G Sn acts the set X 142,3 n'T in the natural way For TE Sm and ie X a i Oci We'll that this is an action verify by L f z checking properties L e i e i i ie V for all X 2 Given a te Sn and EE X P T.i 8 Tci r Tc i rot i roc E V R acts on Ex2 The group G Gln the I set X R by matrix vector multiplication c For A Glen and XER A x Ax have We that 1 I x Ix all ne V for X 2 Given A Be Glen and KEIR A B x A Bx CAB x AB x V Ex3 The G D acts on the group set 3,4 which we can think 4 11,2 of as the set vertices a of of square 1 2 4 3 Here we have that Z z 1 1 v 3 3 4 4 So V 2 1 We could also think of X as the set of the edges of square 1 4 23 In this case I 1 V 2 7 4 2 3 43 So V 2 4 and I L Ex 4 A G acts on itself left group by multiplication Given a be a b ab G an exercise is an As prove that this action Ex 5 A G acts on itself group by t Given a be G a b aba conjugation To see that this is an action note that e a 1 eae a for all AEG 2 a b a Cbgb g 1 t a a Cbgb ab ab ab g g Thus this is indeed an action It will sometimes be to view an helpful action GTX in a different way Suppose that G is a on group acting a set X For G consider ge fixed the map X X 4g x I se g We can show that is a on Hg bijection Indeed see X we have X given that t.se a 4g g g g x gg e ze R so is To see that 4g surjective 4g u is let X Then injective ye Igcse a Ugly g g y i a g g y g g a g g g g y e se e Y se y so is Thus is 4g injective belongs to bijective Consequently 4g the set X is S f X f bijective Exercise For set X the set any as defined above is a under Sx group Lcs composition Moreover X n the if E Sx Sn with this in mind consider once again the action G on X of Define E G Six l 4g g This OI is a between In map groups it is a homomorphism fact group Indeed let c We claim that g g G o Ecg ga Ecg Ilga That is we claim that Ygga Vg Uga a But of course for EX a x Yg g gigs x g gz g Ygate se as 04g 4g 4g g we so Thus have 4g g Vge Yg 01g Eg That is I is g Iga a homomorphism Moral action GTX Every gives rise to a homomorphism G Sx i s g 4g where X X 4g se I goose this is to one knowing fact enough prove the coolest results in the course of Theorem 8 I Cayley's Theorem G is isomorphic to a of Every group group In G he permutations particular if is a then G isomorphic to subgroup of Sh Proof Let G act on itself by left consider homomorphism multiplication and the Of G described above Sq Let the First Isomorphism Theorem By im G I E E Koot Sq is What turd If geker I then 01g is the of Thus 4g identity Sg a a for all AEG a 4g g ga cancellation e so Ker Ie By g o is to a G isomorphic subgroup of Sq Krag 8.2 Orbits 8 Stabilizers Given a action GAX and group see an element G there are a few natural one can ask questions E Where in X can u be sent ii elements fine u which of G Thus we make the definitions following a Definition Let Gnu X be group action Given see X define i the orbit se be the of to set On lg.se gee ii the stabilizer n to be the set of stab G a se Ige g Note that the orbit On is a subset of X while the stabilizer Staba is a subset G In of fact Proposition 8.2 is an action If GTX then for any see X Stab E G Proof Exercise Is a X No On subgroup of In X is not even a general group e C't Ex The group G feio Oer acts on the set X c by multiplication O e For 2 k e Z e'Oz The orbit ZEC is of Oz Z Oz e Oz OER WE E w Z 0 Note that stab eio eio 0 G 0 z and 2 to Stabz e ei0z 42 for usual action Ex Consider the of Sin on on X 11,2 in I Nci ie we can send Given X in to any any c r Thus X ij Oi X g using Stab Oci i By definition lot Sn I Sn t n Note that Oi n Stabi t Ex Let be the all G group of rotations cube of the We will consider the action of G on the cube's vertices edges and faces g L X vertices of the cube Given vertex VE X we can move V to other vertex a rotation using any Or 8 The rotations that leave only V unchanged e are and the rotations about the line V and through its vertex opposite i Stab 3 z edges the cube X of Given E E X we can move E edge to other a rotation edge using any i OE 12 The rotations that leave E only unchanged are e and rotation the about the line and through E its opposite edge E or a Stabe 2 3 faces the cube X of Given Fe X we can move face to other face a rotation any using i Of 6 The only rotations that leave F unchanged are e and the rotations about the line F and its through opposite face F u i Stab 4 Notice that in all cases Staba On 24 It turns out that this value represents the order the G in of group and fact this occurs for every action GAX Theorem 8.3 Orbit Stabilizer Let GAX be For see a group action every X G Stab On In particular if G then G Stab On Proof Define 4 G stab On by 41g Staba g se We claim that 4 is bijective hence G Staba On First we have that for all g gre G g Stabs ga Staba gig C Stabs gig se se g se gz K 41g Uga Thus 4 is well defined 8 injective Moreover given ye On we can write y g x for some gear Then 4 gstaba g x y so 9 is surjective i 4 is bijective as claimed The final claim for finite groups follows from Lagrange's Theorem agama The Orbit Stabilizer Theorem can reveal lots about a group action GAX or about the group G itself Ex What are the possible group actions of G 25 on X 142,33 Well a we have given any X that 25 Staba On so 5 Staba On Hence On L or 5 But since On E3 it must be that On L for all n That is see g a r for all GE 25 and all X The only action is the trivial action Ex How does many rotational symmetries a soccer ball have all such Let G be the group of symmetries and let G act on the set X of all black pentagonal faces of the soccer ball Fix any face FEX This face can be rotated to other such so any face OF 12 Moreover the only rotations that leave F unchanged are the 5 rotations about the axis through F 2 5 Consequently Stab By the Orbit Stabilizer Theorem G Stab OF 5 12 GO Further Applications Burnside's Lemma We will now see how actions can be group to used solve some neat counting problems Ex 1 How different can we make many ways 2 2 chess board black and white a using squares Here are some examples etc wi em I 2 3 4 5 Hold on some of these boards are really the same For instance 2 can be rotated to 3 So We should consider two boards to be the same if one can be rotated into the other That is our be restated as problem may follows If G is the group of rotations of a is 16 square and X the set of all 24 2 2 chess boards how orbits does many the action GAX have Burnside's Lemma us a to count gives way these orbits efficiently First We'll need the following proposition Let be a Proposition 8.4 GAX group action The orbits of the action partition X That is a X On b if x yell then Ose Oy or Oan Oy 0 Proof Assignment 5 Lemma 8.5 Burnside Let G be a finite on a set X If N group acting finite is the number of orbits then N Fixlg g g where for geG Fix g IneX goose se Proof Let n be the number of pairs a c such that se se g Gx X g First note that for a fixed geG the number such is so of pairs gin Fix g n I Fix g gEG Also note that for fixed a X the number of such pairs g n is staba so n stab G l Go I k f sec On But we have so for any rye Or Oy On I t t t 1 yeon Oy On On On On times So I N number of orbits reX On We conclude that I Fixg n G N gea So as claimed N Fix g gaza g g a We can now attempt to solve our chess board problem We wish to determine the number of orbits of all boards under the group G of rotations of a square By Burnside we must find FixLg for all rotations gear Note that G e Rao Riso 12270 g e Rao Riso 12220 2 22 2 Fix g 24 Thus the number of distinct boards orbits is N Fix Ig Egfr g L 16 2 4 2 6 4 Boat atheroma How necklaces Ex many 6 bead can be and 3 made using 3 black beads white beads Solution We can choose the location of 3 beads Es 20 the black in ways and the beads must be white remaining So let X be the set of these 20 possible necklaces and let G D6 be the a We symmetry group of hexagon consider 2 necklaces in X to be the same to the same orbit if they belong under the action Gnr X What are the symmetries in G Rotations V C R R2 Rs R Rs F Fy Fr Flips Fsf Fl Fz Fz Fy E5 Foo F6 We now Fix each compute g for gEG g e R R R R Rs F Fa F F Fo Foo 20 O 2 O 2 O O O O I 22 22 Fix g By Burnside's Lemma the number of orbits i.e the number of necklaces is Zoot 2 2 22 2722 1g Fix g plz g L 36 3 12 They are 1 can one label the Ex How many ways sides a G sided die the of using each of the numbers 1 G exactly once Solution There are 6 ways to put the numbers on but