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 27: Proof of the Orbit-Stabilizer theorem

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

(a) The orbits partition S.
(b) If sS and H=Stab(s) then φ: GH Gs gH gs is a function and φ is a bijection.

Let G be a group and let N be a subgroup of G.

(a) The cosets in GN partition G.
(b) All cosets have the same size.

Idea of proof.

Let gG.
To show: φ: gN N gn n is a function and φ is a bijection.

Let be an equivalence relation on a set S.

The equivalence classes partition S.

Proof of the first Proposition.

To show: The orbits partition S.
To show:
(aa) sSGs=S.
(bb) If s1,s2S and Gs1Gs2 then Gs1=Gs2.
To show:
(aaa) sSGsS.
(aab) SsSGs.
(aaa) Since GsS then sSGsS.
(aab) To show: If aS then asSGs.
Since aS, and aGa, then asSGs.
So SsSGs.
(ab) Assume s1,s2S and Gs1Gs2.
To show: Gs1=Gs2.
Since Gs1Gs2 there exists tGs1Gs2.
So there exist g1,g2G such that g1s1=t=g2s2. So s1=g1-1g2s2 and s2=g2-1g1s1.
To show:
(aba) Gs1Gs2.
(abb) Gs2Gs1.
(aba) To show: If Gs1 then Gs2.
Assume Gs1.
Then there exists hG such that =hs1.
So =hs1=hg1-1g2s2Gs2, since hg1-1g2G.
So Gs1Gs2.
(abb) To show: If mGs2 then mGs1.
Assume mGs2.
Then there exists kG such that m=ks2.
So m=ks2=kg2-1g1s1Gs1, since kg2-1g1G.
So Gs2Gs1.
So Gs1=Gs2.
So the orbits partition G.
To show:
(ba) φ: GH Gs gH gs is a function.
(bb) φ is a bijection.
(ba) To show: If g1H,g2HGH and g1H=g2H then φ(g1H)=φ(g2H).
Assume g1,g2G and g1H=g2H.
Then g1g2H.
So there exists hH with g1=g2h.
To show: φ(g1H)=φ(g2H).
To show: g1s=g2s. g1s= g2hs=g2s, since hStab(s).
To show: φ is a bijection.
To show: ψ: Gs GH t gH where gG such that t=gs is an inverse function to φ.
To show:
(bba) If g1,g2G and g1s=g2s then ψ(g1s)=ψ(g2s).
(bbb) φψ=idGH and ψφ=idGs.
(bba) Assume g1,g2G and g1s=g2s.
Then g1-1g2s=s, so that g1-1g2Stab(s).
To show: ψ(g1s)=ψ(g2s).
To show: g1H=g2H.
To show:
(bbaa) g1Hg2H.
(bbab) g2Hg1H.
(bbaa) To show: If xg1H then xg2H.
Assume xg1H.
Then there exists hH such that x=g1h.
To show: xg2h. x=g1h=g2 g2-1g1h g2H, since g2-1g1Stab(s)=H.
So g1Hg2H.
(bbab) To show: If yg2H then yg1H.
Assume yg2H.
Then there exists kH such that y=g2k.
So y=g2k=g1g1-1g2k g1H, since g1-1g2Stab(s)=H.
So g2Hg1H.
So g2H=g1H.
(bbb) To show: φψ=idGH and ψφ=idGs.
If gG then (φψ)(gH)= φ(ψ(gH))= φ(gs)=gH and (ψφ)(gs)= ψ(φ(gs))= ψ(gH)=gs. So φψ=idGH and ψφ=idGs.

Notes and References

These are a typed copy of Lecture 27 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on October 7, 2011.

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