Group Theory and Linear Algebra

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

Last updated: 24 September 2014

Lecture 25: Group actions, orbits, stabilizers

The dihedral group Dn is the set Dn= { 1,x,x2,, xn-1,y, xy,x2y,, xn-1y } with operation given by (xiyj) (xky)= x(i-kmodn) y(j+mod2) so that, in particular y2=1,xn=1 andyx=x-1y.

A G-set S, or an action of a group G on a set S is a function G×S S (g,s) gs such that

(a) If g1,g2G and sS then g1(g2s)=(g1g2)s,
(b) If sS then 1·s=s.

Let sS. The stabilizer of s is Gs= {gG|gs=s}. The orbit of s is Gs={gs|gG}.

Let G be a group and let S be a G-set.

(a) The orbits of G acting on S partition S.
(b) Let sS. Then Gs is a subgroup of G, φ: GGs Gs gGs gs is a well defined function and φ is a bijection.

"D4 acting on a square". e1 e2 e3 e4 c d b a b c a d a b d c d a c b 1 x x2 x3 b a c d c b d a d c a b a d b c y xy x2y x3y D4 acts on the vertices V={a,b,c,d}: xa=b, xya=a. D4 acts on the edges E={e1,e2,e3,e4}: x2e1=e3, ye1=e1.

"G acts on itself by conjugation". G×G G (g,s) gsg-1 or g·s=gsg-1, where gG and sG.

Let sG. The centraliser of s in G is the stabiliser of s, 𝒵G(s)=Stab(s)= {gG|gsg-1=s}. The conjugacy class of s in G is the orbit of s, 𝒞s={gsg-1|gG}. The centre of G is 𝒵(G)= {sG|gs=sgfor allgG}.

HW: Show that s𝒵(G) if and only if 𝒵G(s)=G.

HW: Show that s𝒵(G) if and only if 𝒞s={s}.

D4 acts on itself by conjugation. x 1 y and x x2 y x x yy x3 x since yxy-1 = x3yy-1=x3 yx2y-1 = x2yy-1=x2 yx3y-1 = xyy-1=x y y xx x2y y since xyx-1= xyx3=xxy= x2y xy x,yy,x x3y So the conjugacy classes are {1}, {x2}, {x,x3}, {y,x2y}, {xy,x3y} and the centralizers of elements in D4 are 𝒵D4(1) = Stab(1)=D4 𝒵D4(x) = Stab(x)= {1,x,x2,x3} 𝒵D4(x2) = Stab(x2)=D4 𝒵D4(x3) = Stab(x3)= {1,x,x2,x3} 𝒵D4(y) = Stab(y)= {1,x2,y,x2y} 𝒵D4(xy) = Stab(xy)= {1,x2,xy,x3y} 𝒵D4(x2y) = Stab(x2y)= {1,x2,y,x2y} 𝒵D4(x3y) = Stab(x3y)= {1,x2,xy,x3y}

Notes and References

These are a typed copy of Lecture 25 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on October 4, 2011.

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